View source for Talk:Prime-counting function

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== Early thread(s) ==
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This is a very strange stub. The format does not fit very well with other math articles, the style is rather poor, and there is an accuracy issue as well (this is what has prevented me from expanding the stub). If I am reading everything correctly, the "recent" discovery is a modified/obfuscated [http://mathworld.wolfram.com/LegendresFormula.html Legendre-Meissel recursion] (we're just counting composite numbers by inclusion-exclusion). Suggestions?

[[User:Alodyne|Alodyne]] 01:06, 6 Dec 2004 (UTC)

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Edited to handle points of dispute.
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This unusual and not terribly effective formula is the work of sci.math denizen James S. Harris, who has produced a large body of work of dubious quality. This morning, he wrote (on sci.math):

''As an experiment I put my prime counting function on the Wikipedia writing the entire article.  You could see how a concise prime counting function fits nicely in an encyclopedia artice, and I saw it as an opportunity to see if the math world could behave at all like expected.''

The usual dispute with James Harris is over the quality and relevance of his work. The quality or relevance is, however, quite irrelevant to its appropriateness here, since its the manner of its appearance is directly contrary to Wikipedia policies (specifically, the policy against self-promotion).

That said, having a page on prime-counting functions (with, for instance, Legendre's method) is a good idea. An internal link to the Prime Number Theorem would be worth having, but redirecting seems misleading, since the Prime Number Theorem provides only an approximate count.
--[[User:Jake Wildstrom|Jake]] 18:18, Feb 1, 2005 (UTC)

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Hey, I have an unusual situation in that mathematicians are refusing to acknowledge a correct and interesting mathematical result, which also happens to fit easily into a relatively small encyclopedia article.  Notice the disparagement in Wildstrom's comments, when the formula itself is just math.

I emphasize that Wildstrom felt it necessary to disparage.  I'm not dealing with people who are behaving objectively from the math world, but they are very emotional on this subject and refusing to just acknowledge a correct mathematical result.  It's personal for me as the discoverer, but it remains to be seen why it's so personal to them.

I have what is an actual prime counting function.  It is easily derived, but I derived it, and apparently mathematicians have decided for that reason they will never acknowledge it.

I plead a special case here as look at the current article and consider how abstruse it is, versus my article.

Also note that no one not an expert in the field could actually count primes with the current article, but my formula allows a non-expert in the field to actually do so, plus I give the additional information about the prime number theorem and current research.

I say that mathematicians are just being weird here, and obviously, if mathematicians refuse to acknowledge a correct and easily proven result, then I run into a problem putting up a citation!

[[User:James Harris|JSH]] 18:56, 5 Feb 2005 (UTC)

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I've added a normal sort of prime-counting function page, which I may expand on. 
[[User:Gene Ward Smith|Gene Ward Smith]] 02:26, 3 Feb 2005 (UTC)

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Consider, the page was reverted back to the redirect and then there was no prime counting function article for a while, until after I mentioned my experiment on the sci.math newsgroup THEN someone came to try and write an article versus just leaving the re-direct to the [[prime number theorem]].

Now compare their abstruse, hard to understand [[prime counting function]] article, which does not contain a formula that a non-expert can use, with my own, and the real issue here I think is a bizarre case of mathematicians fighting to not acknowledge a correct result.

The rules limiting to cited research don't handle a situation where the experts are not behaving as expected with an easily derived math formula, where correctness is not in doubt.

[[User:James Harris|JSH]] 18:56, 5 Feb 2005 (UTC)

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I think the article needs to have the methods of getting an exact count on prime numbers (other than a brute-force sieve), or either a link to a page that has that information.
[[User:Bubba73|Bubba73]] 22:05, 21 Jun 2005 (UTC)

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Most of the methods for counting prime numbers, especially the ones actually used, are too involved for an encyclopedia article.  My prime counting function is distinctive in that it's short enough for an article, while it is slow, though a fast algorithm derived from it is even smaller, so it is small enough with a fast algorithm to fit in an article.

But my research isn't published in a math journal, so I'm not supposed to put it up.  And don't think I could put it up anyway without people coming out of the woodworks to tear it back down as math people are VERY active in fighting my research, mostly relying on personal attacks, like you can see earlier on this talk page.

The real math field is nothing like what many of you probably think it's like, as people are very committed to promoting their careers and academics see amateurs like myself, especially in very high profile areas like prime numbers as not helpful to their own careers.

For instance, I had a paper on a different subject though still number theory published in a peer reviewed electronic math journal and people from sci.math got together and mounted an email campaign against it and the journal folded, pulling my paper, which they could try to do being an electronic journal, though it is unheard of.

So I have the message loud and clear from the mainstream math world which is they will not accept my work despite it being correct, and the rest of the world does not matter.

It's my math, as far as they're concerned, and what's my math is to be rejected without regard to its mathematical importance.

Of course, it's annoying to me, but the world pays the price, though someday I guess, as has happened with past research fought by some group or other it will be "discovered" and I'll just be another story to add to a long list of people with major results who had to fight against the established "experts" who saw their efforts as a threat.

Nothing changes in this area.  People don't learn from history.

[[User:James Harris|JSH]] 21:21, 27 August 2005 (UTC)


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What amazes me is how easily math people get away with dumb crap, like writing an article on counting prime numbers that doesn't actually show you known methods to count prime numbers.

The authors of this article just glancingly mention "arithmetic" methods only go off on a tangent talking about work related to the Riemann Hypothesis.

For those who want to see actual "arithmetic" prime counting formulas see the much better article at MathWorld:

http://mathworld.wolfram.com/PrimeCountingFunction.html

I've sat and waited and waited, hoping that someone from the math community would step up and do the service of writing a real article on prime counting for the Wikipedia, but the math community doesn't care.

Compare the MathWorld article to Wikipedia and see what I mean.


[[User:James Harris|JSH]] 04:10, 13 September 2005 (UTC)


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Oh, and hey, I think it's worth it to compare both articles to my own reverted article:

http://en.wikipedia.org/w/index.php?title=Prime_counting_function&oldid=9142249

My article also talked about figures from the history of prime counting, like Gauss and Chebyshev while the current Wikipedia article acts as if prime counting started in the late 1800's with Riemann!

It's bizarre how bad that article is--like, not even mentioning Gauss??!!!--and how the community here is tolerating that, especially given that it was a reaction article from Usenet, when I talked about having written the first prime counting function article for Wikipedia, as before there was a re-direct.

My guess is that some college students fixated on the Riemann Hypothesis as it's the big thing now wrote the article, and being kids they don't appreciate, or maybe even don't know the rich history in this area, which the interested reader can get more of at the MathWorld article, or hey, at mine.

And them not mentioning Euler or Chebyshev is just amazing to me, as Euler is the one who presented the zeta function, and Chebyshev is the one who did the early limit work that others built upon, so these students--assuming they were--might not have been doing in-depth studies OR their teachers aren't doing their jobs.


[[User:James Harris|JSH]] 15:43, 25 September 2005 (UTC)

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And finally the article has been updated!!!  So a lot of my past criticisms no longer apply.

Intriguingly though, some of the updates include an interesting sieve function which is, it turns out, very closely related to my own prime counting function, and looks a lot like one key piece of it.

In any event, I'm just happy a fuller article was finally written.


[[User:James Harris|JSH]] 21:31, 13 November 2005 (UTC)

== Riemann's counting function ==

In the section on other counting functions, J(x) is given as
:<math>J(x) = \sum_{n=1}^\infty \pi(x^\frac{1}{n})</math>
shouldn't that be
:<math>J(x) = \sum_{n=1}^\infty \frac{1}{n}\pi(x^\frac{1}{n})</math>
? --[[User:Monguin61|Monguin61]] 01:52, 21 December 2005 (UTC)

== Explicit formula ==

Does anyone know what definition of Li is used in the Riemann-von Mangoldt formula? -- [[User:EJ|EJ]] 20:01, 15 January 2006 (UTC)

:Nevermind, I found it here: [http://www-ian.math.uni-magdeburg.de/home/kunik/prime.pdf]. It is quite a mess, actually. I'll try to fix the article later. -- [[User:EJ|EJ]] 18:02, 16 January 2006 (UTC)


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To me the experiment all along has been to see if information in and of itself has a chance to get picked up and acknowledged.

It does not.  It has to be promoted.  It has to be pushed forward against resistance.

You need to be a politician as well as a discoverer or you're just waiting and hoping until some politician picks up your ideas and promotes them, which is a lesson from history as well.

Ideas don't get picked up because of their inherent value, but their value is only seen by groups as a result of the efforts of some people to promote them.

Politics is a crucial part of the researchers' toolkit.

Moving forward as I look to more social ways to make the information enticing, and to excite the imaginations of people about that information, I want it part of the record that knowledge in and of itself, is not enough.

The reality of getting results known is a political process.  You have to promote, campaign, and figure out some way to excite people about the information you have, no matter how huge it is.

I know how big my find is, and how much ground in prime research it covers, but none of that matters to the simple ability of most people to just not notice.

That's the real lesson here, and I think that ends my experiment.


[[User:James Harris|JSH]] 18:06, 8 April 2006 (UTC)

== Comparison Plot ==

here is a comparison plot (ratios) of n/ln(n),Riemann pcf,actual and the approx found on main page (round[exp(x-A*z+B)])

[https://commons.wikimedia.org/wiki/File:PrimeCountingFunction9.svg]  Comparison plot

Now, a power fit to the '''first 3 significant digits''' provides the following approx:

             #primes ~ 504.47/(logN**1.0434)      [w/ user adding the appropriate power of 10]

Thus, for N=23, the eqn returns 19.1 with  E20 implied

This approx uses data for n>=10^6 and returns a value within 1% of the actual.
:: updating above thru 10^29, #primes ~ 4872/(logN^2.822)      [w/ user adding the appropriate power of 10]
::   and returns value within 4% [[User:Billymac00|Billymac00]] ([[User talk:Billymac00|talk]]) 15:12, 16 August 2022 (UTC)

Realizing that the denominator power likely is tending towards unity, and reworking gives

             #primes ~ 1.022*N/lnN

This approx returns a value within 1% of the actual for n>=10^14.--[[User:Billymac00|Billymac00]] 20:57, 20 June 2006 (UTC)
:: the above estimates primepi(10^27) as 16438850389078643254053843.  However, a completely different approach (3/21/2015) estimates 16921932166972187992181811.
Both are within the lower and upper bounds available at [http://mathworld.wolfram.com/PrimeCountingFunction.html 1]

== Questioning controversy ==


With [[natural numbers]] x and n, where <math>p_i</math> is the i_th prime then


:<math>P(x,n) = x - 1 -\sum_{i=1}^n {(P(x/p_i,i-1) - (i-1))}</math>

where if n is greater than the count of primes up to and including <math>\sqrt{x}</math> then n is reset to that count.

P(x,n), with n equal to the count of primes up and including <math>\sqrt{x}</math> is the count of prime numbers up to and including x and that is just a simple formula that can be related to what was previously known in the field.  More compact than anything else similar in the prime counting area.

The fully mathematicized form of the prime counting function shown above, where you don't find find a list of primes first--because the formula finds the primes on its own--is, if <math>y\le\sqrt{x}</math> then

:<math>P(x,y) = \mathrm{floor}(x) - 1 -\sum_{k=2}^y {((P(x/k,k-1) - P(k-1,\sqrt{k-1}))( P(k,\sqrt{k}) - P(k-1,\sqrt{k-1})))}</math>

else <math>P(x,y) = P(x,\sqrt{x})</math>.

Either way gives the same count of prime numbers, of course, but the fully mathematicized form doesn't need to be given a list of primes up to <math>\sqrt{x}</math>, which is how this formula is crucially different from what was previously known in the field.

So you get simplicity one way--compare with prime counting functions on the main page--and complexity another way that does something never seen before, and that should add up to mathematical interest for the math community.

Yes, I've contacted mathematicians including mathematicians leaders in the field in this area, and just any mathematician I could try to get to pay attention and even none mathematicians.  I've also tried to get these results published in a mathematical journal, to no avail.

Turning to the web and Usenet I've gotten ridicule which has been sharp, personal and often cruel with more energy directed the more I question the current math community's behavior, and there seems to be no way to stop them.  It's like they have absolute power and are willing to use it to block whatever suits them to block.

When before I would have assumed that math was just of so much interest to members of that community that they embraced new results, whatever the source.

And you can see similarities to what was previously known but it IS different in key ways.  However, I discovered that formula a few years ago and ran into a wall from the math community simply refusing to properly acknowledge or record it.  They behave like they own math and attack people they see as interlopers, refusing to acknowledge information from outsiders and when you have people breaking rules in such a wacky way, what can you do?

I tried writing a page that showed the more complex form of that equation, which is the fully mathematicized version and you can see at the top of this discussion page how well that went over, but hey, I'm desperate.  Math people are just saying no, refusing to acknowledge a simple formula that counts primes that is different in special ways from what was previously known and there seems to be nothing I can do about it.
[[User:James Harris|JSH]] 02:37, 8 November 2006 (UTC)

HI..james although i'm not a teacher i find your article interesting could you give  your mail?? or similar for contact ?? i would like to take a look at your article. --[[User:Karl-H|Karl-H]] 14:46, 4 April 2007 (UTC)

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:James, as your humble archivist, you know I cling to your every word.  So you make it really difficult when you start flinging your words around the internet like this.  Besides, it is clear that this talk page is not an appropriate forum.  It's for discussing editorial decisions regarding the article, not for drumming up support for original research.
:Come on now.  Let's return to our usual settings.  Thanks much. [[User:Phiwum|Phiwum]] 03:26, 7 November 2006 (UTC)


But, as to your formula, my favorite part is the "floor(x)" where x is a natural number.  Shouldn't that be "floor(ceiling(abs(x+1-1)))"?  Or, I guess x would also work since it's a nonnegative integer and floor, ceiling, absolute value and adding and subtracting 1 each all do nothing to it.

== The tetralectic who? ==

An unregistered editor added a link to the [[Tetralectic constant]] page.  That page gives a "famous" result prove in 2006.  The only references are to Yahoo search results, which I did not wade through.  Google comes up empty for both "Tetralectic constant" and "Tetralectic theorem".  Is this a real result?  If so, maybe someone can clean up that very young article.  If not, perhaps a CfD is in order. [[User:Phiwum|Phiwum]] 15:11, 30 November 2006 (UTC)

:"Tetralectic constant" was poor OR and deleted 6 December 2006 after [[Wikipedia:Articles_for_deletion/Tetralectic_constant]]. [[User:PrimeHunter|PrimeHunter]] 11:42, 13 January 2007 (UTC)

== Lehmer and prime counting function ==

I'm not quite sure that this is relevant - but I'll post it anyway. I seem to remember (from reading one of the Berndt/Ramanujan volumes) that Derrick Lehmer's values for <math>\pi(x)</math> were out by unity - because (for some reason best known to himself) Lehmer took unity itself to be prime. Does this have any relevance to his contributions? [[User:Hair Commodore|Hair Commodore]] 19:22, 12 January 2007 (UTC)

:A few yrs ago I chatted with an analytic # theory professor with a PhD from Berkeley, and I asked her about Lehmer: "Is he the guy who always said 1 is prime?" She smiled and said "1 ''is'' prime." Everyone laughed and somebody said 'It's a Berkeley thing." --I sometimes wonder if the analytic # theorists have some reason for wanting 1 prime to make some formulas or theorems in their area work better. Of course, 'regular' # theory wants 1 to be neither prime nor composite to make the [[Fundamental theorem of arithmetic]] work.[[User:Richard L. Peterson|Rich]] 05:49, 15 March 2007 (UTC)

==What about the sum of the primes less than x?==
Does anyone know what work has been done on that? And sum of squares of primes less than x, etc?
(Analogy with divisor functions) Perhaps Wikipedia could have an article on it.[[User:Richard L. Peterson|Rich]] 05:22, 15 March 2007 (UTC)

== Archiving JSH ==

Although the page is not that large, JSH's apparent failure to sign his missives suggest that we move them to a subpage so we can sign them for him.  Sections 0 (most of, anyway), 2, and 4 seem to be JSH's.  I'm not saying that JSH's formulas are ''wrong'', but they're self-published repeating multiple times, and, being 2-d recursive, not particularly notable, even if apparently fairly simple.  Any comments? &mdash; [[User:Arthur Rubin|Arthur Rubin]] |  [[User_talk:Arthur_Rubin|(talk)]] 16:32, 4 April 2007 (UTC)

* someone removed the formula:

:<math> \pi (x) = \sum_{n=1}^{\infty} \frac{\mu (x)}{n}J(x^{1/n}) </math>

In fact this appears in [[http://mathworld.wolfram.com/RiemannPrimeCountingFunction.html]] but with different name (they call it f(x) instead of J(x) ) --[[User:Karl-H|Karl-H]] 18:34, 4 April 2007 (UTC)

:The formula already appeared a few lines further down.  &mdash; [[User:Arthur Rubin|Arthur Rubin]] |  [[User_talk:Arthur_Rubin|(talk)]] 22:51, 4 April 2007 (UTC)
:Nope, I was wrong.  I've re-added, and removed second reference to [[Möbius inversion]].  Sorry about that. &mdash; [[User:Arthur Rubin|Arthur Rubin]] |  [[User_talk:Arthur_Rubin|(talk)]] 22:57, 4 April 2007 (UTC)

==Requested move==
[[User:Michael Hardy|Michael Hardy]] has moved [http://en.wikipedia.org/w/index.php?title=Prime-counting_function&diff=142997065&oldid=142996923] this article from [[Prime counting function]] to [[Prime-counting function]] without prior discussion. I suggest to move it back per [[Wikipedia:Naming conventions#Use common names of persons and things]]. The form without hyphen seems much more common and it's the only form I have found in the online references in the article. Michael Hardy does apparently not dispute that the form with hyphen is less common, but he thinks the hyphen makes the name easier to understand and should be included for that reason. See the discussion at [[User talk:Michael Hardy#Prime counting function]]. [[User:PrimeHunter|PrimeHunter]] 16:34, 7 July 2007 (UTC)

=== discussion copied from user page ===

==== Prime counting function ====

I see "prime counting function" more often than "prime-counting function". I just did a Google scholar search where the former was also more common. I think this name change is controversial and should be discussed first. [[User:PrimeHunter|PrimeHunter]] 00:01, 7 July 2007 (UTC)

: Here's something I said about hyphens on the discussion page of another article I moved:
: 
:: This article's title, '''Guyou hemisphere in a square projecction''', sounded as if it was about something called a '''Guyou hemisphere''' that was in something called a '''square projection'''.  After reading a bit, I realized that was not what was meant.  That's why I added the hyphens.  Yet another case of how hyphens can be magnificently efficient in conveying information, and why it is to be regretted that their use is often no longer taught. [[User:Michael Hardy|Michael Hardy]] 12:56, 5 July 2007 (UTC)
: 
: That's also the only reason the hyphen was omitted in this case. [[User:Michael Hardy|Michael Hardy]] 00:09, 7 July 2007 (UTC)
::Editors could often argue a lot about what is the "best" name by their own preference. I think we should follow [[Wikipedia:Naming conventions#Use common names of persons and things]]. I just looked at the online references in [[prime-counting function]] and didn't find a single hyphen. Wikipedia is here to report what sources say and not to promote an agenda - not even an apparantly harmless one like increasing the use of hyphens. [[User:PrimeHunter|PrimeHunter]] 00:34, 7 July 2007 (UTC)

Hyphens are something everyone's accustomed to seeing since they're still used in the traditional way in newspapers, magazines, and books, and this particular hyphen, like many hyphens, aids understanding.  "Prime counting funnction" could be taken by the reader to mean one of many "counting functions", singled out as the "prime" one.  But that is not what is meant.  The hyphen makes that clear. [[User:Michael Hardy|Michael Hardy]] 15:17, 7 July 2007 (UTC)

:I have suggested to move back at [[Talk:Prime-counting function#Requested move]]. [[User:PrimeHunter|PrimeHunter]] 16:35, 7 July 2007 (UTC)

=== end of discussion copied from user page; continue below ===

== Table of π(x), x / ln x, and Li(x) ==

The significance of the ''differences'' between π(x) and x / ln x, and π(x) and Li(x), depends on the magnitude of the values, and as the numbers get large it's increasingly difficult to tell by eye how significant these differences actually are. Suggest replacing with a table that in some suitable way shows the ''relative'' error. This would be more useful for getting a quick idea of how accurate the approximations are. Matt 19:55, 2 January 2008 (UTC).

I don't understand the relative error columns. For example x = 10, pi(x) - x / log(x) = -0.34, which rounds to 0. The relative error is shown as 8.22%. If that is based on the rounded difference it should be 0.00%; if based on the unrounded value it should be -0.34 / 4 * 100% = -8.57%. [[User:HannsEwald|HannsEwald]] ([[User talk:HannsEwald|talk]]) 21:40, 7 August 2024 (UTC)

==== Prime Counting Function With arctan ====

About 3/4 of the way down the page is a formula for pi(x) that has an arctan in it.  Is this formula true?  I cannot find it in the literature.  Does anyone have a reference for it?  <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:MathPerson|MathPerson]] ([[User talk:MathPerson|talk]] • [[Special:Contributions/MathPerson|contribs]]) 18:39, 21 April 2010 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

== Might be useful? ==
the mobius numbers satisfy sum mu(n)/n=0 and this explains why the literature defines the R-function with either li
or Li  <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/2A00:1620:C0:64:21C:61FF:FE03:A4C|2A00:1620:C0:64:21C:61FF:FE03:A4C]] ([[User talk:2A00:1620:C0:64:21C:61FF:FE03:A4C|talk]]) 07:49, 16 December 2012 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

== 'log' and 'ln' ==

I don't think the decimal logarithm is being used at all in this article, so somebody with time should--if s/he knows the wikipedia standard--make all uses here one or the other.[[Special:Contributions/204.108.237.194|204.108.237.194]] ([[User talk:204.108.237.194|talk]]) 20:09, 22 December 2012 (UTC)

Standardizing Wikipedia is a very ambitious goal. Clarifying an individual article is much easier, though, which is why I added the text "where log is the [[natural logarithm]]" to this article.[[User:DubleH|DubleH]] ([[User talk:DubleH|talk]]) 12:15, 13 August 2022 (UTC)

== Better prime counting formula ==

a better basic prime counting formula is proposed as follows:

<math> \pi(x)=\alpha.x^{\beta} \quad \textrm{with:} \quad \alpha=0.0777498457 \quad and \quad \beta=0.9732770961</math>

The error in this calculation ranges mostly between -4% to +4% for x below 1E+25 and [http://www.slideshare.net/ChrisDeCorte1/better-prime-counting-formula seems] to be better than the pure version of the [http://en.wikipedia.org/wiki/Logarithmic_integral Logarithmic Integral] x/lnx.

[[User:Chrisdecorte|Chrisdecorte]] ([[User talk:Chrisdecorte|talk]]) 15:40, 11 September 2013 (UTC)

:This is [[WP:OR|original research]] so it doesn't belong in the article. Anyway, over large ranges your approximation is far worse than x/ln(x).
:The [[prime number theorem]] says <math>\lim_{x\to\infty}\frac{\pi(x)}{x/\ln(x)}=1</math>.
:Your approximation has <math>\lim_{x\to\infty}\frac{\pi(x)}{\alpha.x^{\beta}}=\infty</math>.
:For a specific small range you can choose α and β to match that range OK, but it always breaks down over large ranges. π(x) simply doesn't grow like that. This is a proven fact so it's no use trying to find the "right" α and β. At http://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/25295 I gave data computed by [[PARI/GP]] for other α and β suggested by you. Here it is for your above values:
<pre>
// http://oeis.org/A006880 Number of primes < 10^n
{A006880=[4,25,168,1229,9592,78498,664579,5761455,50847534,
455052511,4118054813,37607912018,346065536839,3204941750802,
29844570422669,279238341033925,2623557157654233,24739954287740860,
234057667276344607,2220819602560918840,21127269486018731928,
201467286689315906290,1925320391606803968923,
18435599767349200867866,176846309399143769411680];}

c(x)=0.0777498457*x^0.9732770961;
\p 5
print("x, pi(x)/c(x), pi(x)/(x/log(x)), pi(x)/(x/(log(x)-1))");
for(n=1,#A006880,x=10^n;p=A006880[n];\
print("10^"n" "p/c(x)" "p/(x/log(x))" "p/(x/(log(x)-1))));

x, pi(x)/c(x), pi(x)/(x/log(x)), pi(x)/(x/(log(x)-1))
10^1 5.4712 0.92103 0.52103
10^2 3.6365 1.1513 0.90129
10^3 2.5988 1.1605 0.99250
10^4 2.0218 1.1320 1.0091
10^5 1.6781 1.1043 1.0084
10^6 1.4605 1.0845 1.0060
10^7 1.3149 1.0712 1.0047
10^8 1.2123 1.0613 1.0037
10^9 1.1378 1.0537 1.0029
10^10 1.0829 1.0478 1.0023
10^11 1.0422 1.0430 1.0019
10^12 1.0122 1.0391 1.0015
10^13 0.99050 1.0359 1.0013
10^14 0.97553 1.0332 1.0011
10^15 0.96607 1.0308 1.0010
10^16 0.96126 1.0288 1.0008
10^17 0.96046 1.0270 1.0007
10^18 0.96319 1.0254 1.0006
10^19 0.96908 1.0240 1.0006
10^20 0.97785 1.0227 1.0005
10^21 0.98929 1.0216 1.0005
10^22 1.0032 1.0206 1.0004
10^23 1.0196 1.0196 1.0004
10^24 1.0383 1.0188 1.0004
10^25 1.0592 1.0180 1.0003
</pre>
:All you have done with the new α and β is change the small range where the approximation is OK. Now it is around 10<sup>12</sup> to 10<sup>13</sup>. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 23:33, 11 September 2013 (UTC)

I checked it again for these bigger numbers and your remark might be correct.
Therefore I temporarily suggest following improved formula:

<math>\pi(x)={\alpha.x^{\beta}.10^{(\gamma-x)}+x/\ln x\over 1+10^{(\gamma-x)}} \quad \textrm{with:} \quad \alpha=0.2083666 \quad \beta=0.9294465 \quad \gamma=1E+10 </math>

[[User:Chrisdecorte|Chrisdecorte]] ([[User talk:Chrisdecorte|talk]]) 03:11, 13 September 2013 (UTC)
:Still no. Fitting polynomials is the wrong thing to do here, and by making your guesswork more complicated you're merely hiding its incorrectness rather than making actual progress. The error in the x/log x approximation is known and does not have the form you show. Rather, there is a small error (but still larger than your correction terms) coming from approximating the prime-counting function by the [[logarithmic integral]] (see the History section of this article) and larger error terms coming from the approximation of the logarithmic integral by x/log x: see [[Logarithmic integral function#Asymptotic expansion]]. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 18:10, 13 September 2013 (UTC)

::Yes, and I wonder whether you really meant your expression to contain 10<sup>10000000000−x</sup>.  That's not practical. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 18:19, 13 September 2013 (UTC)

I really appreciate your feedback and I could come back with a new attempt replacing x/lnx in my last formula with 10^(0.9732770526*LOG10(x)-1.10929958) which is the linear approach for values between 1E+10 and 1E+25 and which is better than x/lnx in 7 out of the 16 examples. However, who will tell how things will progress (far) after 1E+25??? Actually, what puzzles me most is how Riemann could find the expression x/lnx ... [[User:Chrisdecorte|Chrisdecorte]] ([[User talk:Chrisdecorte|talk]]) 19:32, 13 September 2013 (UTC)

:As demonstrated above, x/(log(x)-1) is a good improvement over x/log(x), and it's known to remain good: If f(x) is defined as the function which makes π(x) = x/(log(x)-f(x)), then it's proven that f(x) tends to 1. I'm guessing your increasingly complicated expressions are worse than the simple x/(log(x)-1). [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 22:52, 13 September 2013 (UTC)

The f(x) that is talked about above should fit following couples:
x       f(x)
1E+05	1.085196823
1E+10	1.05037
1E+12	1.04087
1E+19	1.02460
1E+25	1.01835

This corresponds with approximations like: f(x)= 1.0886089228x-0.0012909804 or f(x)= -0.0013531392*ln(x) + 1.0881210617 
[[User:Chrisdecorte|Chrisdecorte]] ([[User talk:Chrisdecorte|talk]]) 05:56, 17 September 2013 (UTC)
:Chris and others, could I respectfully suggest that this discussion be taken somewhere other than Wikipedia?  Neither this talk page, nor anywhere else on Wikipedia, is for discussing unpublished mathematical theorems and methods.  There are a number of mailing lists, newsgroups, and web forums where this sort of discussion could continue. —[[User:Psychonaut|Psychonaut]] ([[User talk:Psychonaut|talk]]) 20:31, 17 September 2013 (UTC)

== Unusable But ... ==

from [[Formula for primes]] Talk, someone posted this eq'n, which quickly is unruly but it seems to work.  I show in the format for Pari:
PrimePi(n) = 2+sum(x=5,n,(floor( (x!/x^2)-floor(x!/x^2)+0.3 ))).  --[[User:Billymac00|Billymac00]] ([[User talk:Billymac00|talk]]) 23:46, 29 July 2014 (UTC)

== Request to add a li based formula. ==

I request that the formula that was removed be put back.  In the words of user Sapphorain: "the proposed formula was just an imprecise reformulation of the formula for Pi_0 just above".  In that case, a formal citation should not be needed.

I propose to add (right below "An extensive table of the values of Δ(x) is available."):

"""There is also a logarithmic integral based formula for the prime counting function.
:<math>\pi(x) = li(x) - log(2) - \sum_{k=2} \frac{\pi(x^{1/k})}{k} - \sum_\rho li(x^\rho)+\int_x^\infty dt/(t*(t^2-1)*log(t))</math>

This formula is interesting with its recursive pi(x) calls (requiring negligible computation), and has only one li(x) term per zeta zero.  It may be better suited for computer calculations. The final term can be included for correctness, but is always smaller than 1/(2*x^2*log(x)), and can be removed in calculations because of its small size."""

[[User:NumberCurious|NumberCurious]] ([[User talk:NumberCurious|talk]]) 00:45, 16 June 2016 (UTC)
:Most of the things you want to say about this formula ("interesting"..."negligible computation"..."better suited for computer calculations"..."always smaller and can be removed") require [[WP:RS|reliable sources]]. And if we omit saying these things (because they're unsourced) there isn't much left of interest. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 00:57, 16 June 2016 (UTC)

Sources: [http://trac.sagemath.org/attachment/ticket/8135/riemann_pi.gp riemann_pi.gp] (original publishing of the formula), [http://wstein.org/edu/2010/414/projects/stueve.pdf paper using formula] (page 8 in particular, which demonstrates the correctness of the formula), and the (July, August, September 2014) addition to the Google code project "primecounter", which is a complete reprogramming of the formula, with optional modifications and error tables.</br>
How shall we proceed?</br>
[[User:NumberCurious|NumberCurious]] ([[User talk:NumberCurious|talk]]) 04:36, 16 June 2016 (UTC)
:We could start by you telling us the peer-reviewed mathematics journals those were published in. Because otherwise they don't look reliable for this material. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 05:38, 16 June 2016 (UTC)

It was stated above that the proposed formula is a reformulation of a formula already on the page (we don't need a peer-reviewed mathematics journal for a single mathematical step).  The formula is interesting in that it is [[logarithmic integral | "logarithmic integral]] based", not requiring the Riemann prime counting function or the Gram series.  A series or asymptotic expansion can be used (both on the [[logarithmic integral]] page).  Code has been provided.  Any reviewer can run this code and verify the formula.  The [[four color theorem]] comes to mind, which used exhaustive computer search.  I am stating computer verification of the formula as a source.  I also believe that the page would be better with the formula than without.

[[User:NumberCurious|NumberCurious]] ([[User talk:NumberCurious|talk]]) 18:57, 16 June 2016 (UTC)
: The proposed formula is a trivial (and imprecise) reformulation of the formula just above, i.e.
::<math>\Pi_0(x) \left(= \sum_{n=1}^\infty \frac1n \pi_0(x^{1/n})\right) = \operatorname{li}(x) - \sum_{\rho}\operatorname{li}(x^{\rho}) - \ln 2 + \int_x^\infty \frac{dt}{t(t^2-1) \ln t},</math>
:and there is thus no reason nor interest to mention it in addition. Even with the four colour theorem in mind. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 19:38, 16 June 2016 (UTC)

I believe we have a minor typo, the capital pi, <math>\Pi</math> is being used for the Riemann prime counting function, a lower case pi, <math>\pi</math> is being used for the prime counting function.  We need to change the capital character in the formula to a lowercase letter.  This typo confused me.  I would correct the typo myself right now, but considering the controversy of my proposed addition, I will leave it to someone else to make the correction.  To be precise, the formula with the typo is:</br>
:<math>\Pi_0(x) = \operatorname{li}(x) - \sum_{\rho}\operatorname{li}(x^{\rho}) - \ln 2 + \int_x^\infty \frac{dt}{t(t^2-1) \ln t}.</math></br>which is in the "Formulas for prime-counting functions" section.
[[User:NumberCurious|NumberCurious]] ([[User talk:NumberCurious|talk]]) 00:39, 17 June 2016 (UTC)
:It is not a typo. It is a different function. The distinction between the two different functions given by lowercase and uppercase pi is explained in the section "Other prime-counting functions". —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 00:50, 17 June 2016 (UTC)
I believe that a subtle error is being made.  I have used the formula in two major computing projects, to compute the prime counting function, not Riemann's prime counting function.  Shall I correct the mistake?
[[User:NumberCurious|NumberCurious]] ([[User talk:NumberCurious|talk]]) 18:09, 17 June 2016 (UTC)

==Re-instatement of inequality limits edit==
I have re-instated the edit by [[User:Drriemann|Drriemann]]. His original edit has merit. It was removed by [[User:Sławomir Biały|Sławomir Biały]] because it was primary/self-published source material. If you disagree with my re-instatement please state why the published article in arXiv is not reliable. It looks sound to me. [[User:Fjackson|Frank M Jackson]] ([[User talk:Fjackson|talk]]) 12:26, 25 June 2016 (UTC)

:Arxiv is not a peer-reviewed reliable source.  Furthermore, typically original research articles are primary sources (e.g., for a theorem proven in that spurce), while Wikipedia [[WP:PSTS|should be based on secondary sources]]. [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 12:44, 25 June 2016 (UTC)

::I agree that the their paper in Arxiv appears to be a primary source but we cannot be certain. It may have been publish originally in the maths dept at Bristol University with proper peer-review. Certainly the authors would not be allowed 300 hours of running time on nodes of the University of Bristol’s Bluecrystal cluster if they were not carrying out peer-reviewed research. I have read the article and my view is that the original edit should remain. [[User:Fjackson|Frank M Jackson]] ([[User talk:Fjackson|talk]])

:::So, we agree that it is a primary self-published source.  Our guidelines therefore suggest that the source is not appropriate for the encyclopedia. [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 13:24, 25 June 2016 (UTC)

::::No, I have said that it "appears" to be a primary self-published source and I have given you reason for doubting this. All wikipedia rules are not black and white. These rules should not be applied pedantically. However, if you genuinely believe that the original edit and paper have no merit revert my edit. If you do I will not pursue the matter further but I think that you will diminish the article. [[User:Fjackson|Frank M Jackson]] ([[User talk:Fjackson|talk]]) 13:55, 25 June 2016 (UTC)

{{Od}} It is a primary self-published source, ''by definition''.  There's very little room for interpretation.  True, there are shades of grey sometimes, but this isn't one of those. [[User:Sławomir Biały|<span style="text-shadow:grey 0.3em 0.3em 0.1em; class=texhtml">Sławomir Biały</span>]] ([[User talk:Sławomir Biały|talk]]) 14:04, 25 June 2016 (UTC)
: Arxiv is not at all a reliable source. It is not peer-reviewed (some cranks have been able to publish their complete works on it). If this result is sound and valuable, there is no doubt it will eventually be published in a good peer-reviewed journal, and be reviewed by MathSciNet and Zbl. Then, and only then, it can be mentioned in a wikipedia article. In the meantime it is not proved encyclopedic, must be deleted, and I will delete it now. The argument that it "may have been published originally at the maths dept at Bristol University with proper peer-review" is ridiculous: nobody publishes a paper at Arxiv ''after'' publishing it in a good journal! And even if it were the case, we need a good source to prove ''that'' before we can mention the paper. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 19:58, 26 June 2016 (UTC)

::Gentlemen, I have looked on Google scholar for the paper in question and have found that is has been officially published by Experimental Mathematics, 2015 - Taylor & Francis. The paper has been cited 3 times according to Google Scholar. See
 http://www.tandfonline.com/doi/abs/10.1080/10586458.2014.990118?journalCode=uexm20 
::This goes to show that one should not delete edits from the standpoint that it "appears" to be primary/self-published source material simply because the paper has been downloaded to ArXiv by one of its authors. Of course, I should have investigated further as to which journal it had been published so as to support my original re-instatement.
::Can one of you please re-instate the edit so that the paper referenced in the article is properly cited and can be downloaded? Thanks. [[User:Fjackson|Frank M Jackson]] ([[User talk:Fjackson|talk]]) 19:51, 27 June 2016 (UTC)
::: Yes, it is true that a corrected version of the paper has appeared in Exp. Math 24, 3 (2015), 289-294, and that it has been now reviewed by MathSciNet. So I agree that  the reference to ''this'' version [http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=JOUR&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=Dudek%2C%20A%2A&s5=Exp.%20Math.&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq]  can be reproduced by wikipedia (and ''not'' the ArXiv version: the result is not exactly the same). The paper can of course ''not'' be downloaded, however: this publication is not free, and wikipedia is not a university maths department! [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 22:13, 27 June 2016 (UTC)
::: Regarding the statement that "one should not delete edits from the standpoint that it "appears" to be primary/self-published source material simply because the paper has been downloaded to ArXiv by one of its authors" my point of view is that it is exactly the other way around: it  should on the contrary be deleted without delay unless a proof is provided that it has been published in a good journal. As it happens, the ArXiv version does not appear reliable in this case and thus its mention is not acceptable here. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 22:32, 27 June 2016 (UTC)
:::: I'm not sure I agree with the statement that MR reviews are a necessary and sufficient condition for inclusion (that was also recently made about another one of Drriemann's contributions, which was also a Dudek reference).  Ideally a secondary source should be better than an MR review, like a peer-reviewed scholarly source that discusses the work in context.  MR reviews do not rise to that standard: they are not peer reviewed, and generally do not make evaluative claims.  <small><span style="display:inline-block;vertical-align:-.3em;line-height:.8em;text-align:right;text-shadow:black 1pt 1pt 1pt">[[User:Slawekb|<big>S</big>ławomir]]<br/>[[User talk:Slawekb|<span style="color:red;">Biały</span>]]</span></small> 00:25, 28 June 2016 (UTC)
:::::In many cases MR "reviews" merely copy the abstract of the paper. Only the ones with a listed reviewer, and original text, should be counted as sources at all. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 01:07, 28 June 2016 (UTC)
:::::: Yes, I agree that an MR review all by itself is not sufficient, and that it should not be a reproduction of the abstract (in this case it is original text, by Raul Duran Diaz). Ideally, we should wait until this article is in turn cited as reference in another paper reviewed by MR. Given the nature of the result, which entirely settles under the RH an inequality of Ramanujan, I am convinced that this will eventually be the case, probably very soon. But we can wait until this happens. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 07:17, 28 June 2016 (UTC)

== Link? and new link ==

https://link.springer.com/article/10.1007%2Fs11139-011-9362-6

This is a paper with the that deals with <math>\pi^2(x)</math> Ramanujan formula. 

The link stated with this formula must be the wrong book. Try to check if it should be book four instead of three. I do not have them so I can not.

: The slightly later [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/AMEN/2013/130409(finish).pdf 2013 paper] indicates it is in Berndt part IV.  Indeed, it is in chapter 24, pages 112-113 of that book, shown as "Entry 2 (p. 310)".  It also references the work of Wheeler ("An inequality of Ramanujan") and Keiper/Galway (no direct refernce) doing calculations related to this.  [[User:DAJ NT|DAJ NT]] ([[User talk:DAJ NT|talk]]) 21:08, 27 February 2018 (UTC)

== Recent work of Kristyan ==

I noticed this work  [https://aip.scitation.org/doi/pdf/10.1063/1.5044134 1], and while I cannot speak to the paper's quality, the formula offered seems to work well through the 10^27 entry of the article.  I just make note of it...--[[User:Billymac00|Billymac00]] ([[User talk:Billymac00|talk]]) 00:23, 31 July 2018 (UTC)

: Why not just use the logarithmic integral? I mean, sure, using another term in the asymptotic expansion will help, although it's a little silly to claim that it's counting the twin primes (especially since the next term has coefficient 1, not 2). But sure, whatever approximation works for you! - [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 17:14, 19 November 2021 (UTC)

== Link to this Talk ==
I add a Link to this Talk, because if you searce in German for {{pi}}(''x'') there is only a Link to the German page of [[Prime number theorem]] [[User:Steedmarks|Steedmarks]] ([[User talk:Steedmarks|talk]]) 09:33, 16 October 2018 (UTC)
:{{ping|Steedmarks}} We don't link to talk pages in See also sections so I reverted it. I'm not sure what you mean but this is the English Wikipedia and our links are not determined by Wikipedias in other languages. If you mean there is no link to German under "Languages" then it is because the German Wikipedia does not have a separate article about the prime-counting function. Their article about the [[Prime number theorem]] is linked under "Languages" in that article. Two articles in one Wikipedia language does not link the same article in another language. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 10:09, 16 October 2018 (UTC)

== Exact form ==

According to page 249 of Borwein, et al<ref>{{cite journal |author1 = Jonathan M. Borwein |author-link1 = Jonathan Borwein |author2 = David M. Bradley |author3 = Richard E. Crandall |author-link3 = Richard Crandall |title=Computational strategies for the Riemann zeta function |journal=J. Comp. App. Math. |date=2000 |volume=121 |issue=1–2 |pages=247–296 |url=https://cr.yp.to/bib/2000/borwein.pdf |doi=10.1016/S0377-0427(00)00336-8 }}</ref> the first two formulas involving <math>\pi(x)</math> and <math>R(x)</math> do not have proofs in the literature. As things stand now, the section "Exact form" merely links to an assertion on a Russian website. This doesn't seem to me like a reliable reference.

Also, can someone give a link to the formula that has the arctan function?

There is a lot of good material about exact formulas later on in the section "Formulas for prime-counting functions", including a graph that shows how you can use complex zeros of the zeta function to compute <math>\pi(x)</math>. Maybe the section "Exact form" could be deleted, or merged with "Formulas for prime-counting functions". [[User:MathPerson|MathPerson]] ([[User talk:MathPerson|talk]]) 18:54, 19 May 2019 (UTC)

: The purpose of the "Exact form" section is to note the end-result formula for π(x) in historical context.  As the "Formulas" section states, that form is the Möbius inversion of Π(x), whose form Riemann derived in [[On the Number of Primes Less Than a Given Magnitude]] independently of the Riemann hypothesis.  There are several other sources for Π(x), which "Formulas" references.  I linked that website as a quick fix to guide the curious because it detailed the actual steps of the Möbius inversion; properly someone should find a textbook cite, but this suffices for now.  You may note also that the "Formulas" section lists the simplified form of π(x) involving arctan, and its citations reference Riemann and Von Mangoldt's papers proving that form.
: Nonetheless, some of the top-line results of "Exact form" are missing from "Formulas" and should be copied. —<span style="white-space: nowrap;font-family: Georgia, sans-serif;">[[User:wing gundam|<span style="color: #FF0000">wing</span>]] [[User Talk:wing gundam|<span style="color: #0000FF">gundam</span>]]</span> 20:16, 22 May 2019 (UTC)

{{reflist-talk}}

== Inequalities ==

In the inequalities section, this appears:
:<math> n (\log (n \log n) - 1) < p_n < n {\log (n \log n)} </math> for ''n'' ≥ 6.
The upper bound is sourced to Rosser (1941), and the lower bound to Dusart (1999).  However, further down the section there appear inequalities sourced to Dusart (2010):
: For <math>n \ge 688383</math>,
::<math>p_n \le n \left( \log n + \log \log n - 1 + \frac{\log \log n - 2}{\log n} \right),</math> 
:and (Proposition 6.7) that, for <math>n \ge 3</math>,
::<math>p_n \ge n \left( \log n + \log \log n - 1 + \frac{\log \log n - 2.1}{\log n} \right) .</math>
However, isn't the first of these a strictly weaker upper bound than Rosser's earlier one?  Unless I am missing something obvious (very possible), at presumably Dusart's upper bound from 2010 should be removed from the article.  [[Special:Contributions/164.52.242.130|164.52.242.130]] ([[User talk:164.52.242.130|talk]]) 19:17, 16 November 2021 (UTC)

:: Dusart's bounds are both stronger than their respective bounds above.
:::<math>n\left(\frac{\log \log n - 2.1}{\log n} \right)</math>
:: is positive so his lower bound is larger, making it stronger, and
:::<math>n\left(-1 + \frac{\log \log n - 2}{\log n} \right)</math> 
::is negative, making his upper bound smaller, hence stronger. In general you can see that his bound follows the asymptotic expansion more closely (for more terms) and so at a glance you know it will be stronger. - [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 16:52, 19 November 2021 (UTC)
::: Ah, ok!  [[Special:Contributions/164.52.242.130|164.52.242.130]] ([[User talk:164.52.242.130|talk]]) 12:27, 26 November 2021 (UTC)

== arctan-log terms ==

The sum
:<math>\sum_{n=1}^{\infty} \frac{\mu(n)}{n} \left( - \frac{n}{2 \log x} + \int\limits_{x^{1/n}}^{\infty} \frac{dt}{t (t^2-1) \log t} \right) = \frac{1}{\pi}\arctan\frac{\pi}{\log x}</math>
comes straightly from the expression (32) in paper by Riesel & Göhl.<ref name="RieselGohl">{{Cite journal | author1-link=Hans Riesel | last1=Riesel | first1=Hans | last2=Göhl | first2=Gunnar | title=Some calculations related to Riemann's prime number formula | doi=10.2307/2004630 | mr=0277489  | year=1970 | journal=[[Mathematics of Computation]] | issn=0025-5718 | volume=24 | issue=112 | pages=969–983 | jstor=2004630 | publisher=American Mathematical Society}}</ref>

The sum <math>\sum_{n=1}^{\infty} \frac{\mu(n)}{n} \sum_{\rho}\mathrm{li}(x^{\rho/n})</math> does not converge because

<math>\sum\limits_{\rho}\mathrm{Ei}({\rho}z) = \frac{1}{2z} + \frac32(\gamma+\log z) - \log\sqrt{4\pi} + \frac56 z + O[z^{2}]</math>

for <math>0 < z < \log2</math>, where <math>\gamma</math> is [[Euler's constant]]. Nevertheless, we still write

:<math>\sum_{n=1}^{\infty} \frac{\mu(n)}{n} \sum_{\rho}\mathrm{li}(x^{\rho/n}) = \sum_{\rho}\mathrm{R}(x^{\rho})</math>

in analytical sense since

:<math>\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)}</math>

and actually converging sum is

:<math>\sum_{n=1}^{\infty} \frac{\mu(n)}{n} \left( \sum_{\rho}\mathrm{li}(x^{\rho/n}) - \frac{n}{2 \log x} \right) = \sum_{\rho}\mathrm{R}(x^{\rho}) + \frac{1}{\log x}</math>.

Summarizing this, we have

:<math>\sum_{n=1}^\infty \frac{\mu(n)}n \Pi_0\bigl(x^{1/n}\bigr) = \sum_{n=1}^\infty \frac{\mu(n)}n \left( \mathrm{li}(x^{1/n}) - \sum_{\rho} \mathrm{li}(x^{\rho/n}) - \log 2 + \int_{x^{1/n}}^\infty \frac{dt}{t \left(t^2 - 1\right) \log t} \right) = \mathrm{R}(x) - \sum_{\rho}\mathrm{R}(x^\rho) - \frac{1}{\log{x}} + \frac{1}{\pi} \arctan{\frac{\pi}{\log{x}}}</math>

One could also find <math>\sum\limits_{m=1}^{\infty}R(x^{-2m}) = \frac{1}{\pi}\left(\tfrac{\pi}{\log x} - \arctan\tfrac{\pi}{\log x}\right)</math> directly from
:<math>R(e^{-z}) = \sum\limits_{k=1}^{\infty}\frac{\mu(k)}{\pi}\left(\frac{2\pi}{k z}-\arctan\frac{2\pi}{k z}\right)-\sum\limits_{\rho}\frac{\Gamma(1-\rho)}{\zeta'(\rho)(1-\rho)z^{1-\rho}}</math>
(see [http://www-m3.ma.tum.de/bornemann/RiemannRZero.pdf the paper of Bornemann] for details). [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 11:38, 24 December 2021 (UTC)

:: You say
::"''and actually converging sum is''

::<math>\sum_{n=1}^{\infty} \frac{\mu(n)}{n} \left( \sum_{\rho}\mathrm{li}(x^{\rho/n}) - \frac{n}{2 \log x} \right) = \sum_{\rho}\mathrm{R}(x^{\rho}) + \frac{1}{\log x}</math>."

:: The equality <s>is not correct and</s> does not reflect Riesel and Gohl. And Bornemann's paper does not say anything about <math display="inline">\sum_{m=1}^\infty R(x^{-2m})</math>. I'll have to revert your edits again. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 14:26, 24 December 2021 (UTC)

::: The equality you quoted follows from equalities written before it. Bornemann's paper does say about <math>R(e^{-z})</math>, the rest is pretty obvious for an analytical number theorist. You may ask some of them if you want. [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 17:53, 24 December 2021 (UTC)

::::"''Follows from...''"? From what exactly? How? Please show us your ''original research''.
::::(32) in Riesel and Gohl is
::::<math>\sum_{n=1}^N \frac{\mu (n)}{n}\left(\int_{x^{1/n}}^\infty \frac{dt}{(t^2-1)t\log t}-\log 2\right)=\frac{1}{2\log x}\sum_{n=1}^N \mu (n)+\frac{1}{\pi}\arctan\frac{\pi}{\log x}+\epsilon (x,N)</math>
::::where <math>\epsilon\to 0</math> as <math>N\to\infty</math>. For their computation purposes, Riesel and Gohl chose (page 979) <math>N=154</math> which gives
::::<math>\sum_{n=1}^{154}\frac{\mu (n)}{n}\left(\int_{x^{1/n}}^\infty \frac{dt}{(t^2-1)t\log t}-\log 2\right)=-\frac{1}{\log x}+\frac{1}{\pi}\arctan\frac{\pi}{\log x}+\epsilon (x,154).</math>
::::This does ''not'' imply that
::::<math>\sum_{n=1}^\infty \frac{\mu (n)}{n}\left(\int_{x^{1/n}}^\infty \frac{dt}{(t^2-1)t\log t}-\log 2\right)=-\frac{1}{\log x}+\frac{1}{\pi}\arctan\frac{\pi}{\log x}</math>
::::by any means.
::::The statement "''The arctan-log terms *are* equivalent to the sum over trivial zeta zeros, just look at the expression (32) in the paper by Riesel & Göhl and let N go to infinity.''" in your edit summary is false. Did you perhaps mean something else? [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 19:57, 24 December 2021 (UTC)
:::::You forget to use <math>\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)}</math>. And that's too simple to be called original research. I provided two independent sources to show that equality does hold. That's enough if you understand Möbius inversion in analytical sense. [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 01:52, 25 December 2021 (UTC)
::::::I don't think so.
::::::<math>\sum_{n=1}^\infty \frac{\mu (n)}{n^s}=\frac{1}{\zeta (s)}</math> is valid for <math>s\in\mathbb{C}</math> such that <math>\Re s>1</math>, and if the Riemann hypothesis is true, then <math>\sum_{n=1}^\infty \frac{\mu (n)}{n^s}=\frac{1}{\zeta (s)}</math> is valid for <math>s\in\mathbb{C}</math> such that <math>\Re s> \tfrac{1}{2}</math>. In any case, the equality is not valid for <math>s=0.</math> And none of your sources prove your "result" in any way. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 02:07, 25 December 2021 (UTC)
:::::::The function <math>\sum_{n=1}^\infty \frac{\mu (n)}{n^s}=\frac{1}{\zeta (s)}</math> is analytical as well as <math>\sum_{n=1}^\infty \frac1{n^s}=\zeta (s)</math>. You should understand that series convergence is used differently in the context of analytic continuation. And, again, that's not ''my result'', that's obvious corollary from Riesel&Göhl. Anyway, if you don't like <math>\sum_{n=1}^\infty \mu (n)</math> divergence, you may apply Möbius inversion to
:::::::<math>R(e^{-z}) = \sum\limits_{k=1}^{\infty}\frac{\mu(k)}{\pi}\left(\frac{2\pi}{k z}-\arctan\frac{2\pi}{k z}\right)-\sum\limits_{\rho}\frac{\Gamma(1-\rho)}{\zeta'(\rho)(1-\rho)z^{1-\rho}}</math>
:::::::which follows directly from the Corollary on page 2 in Bornemann's paper (the first term of RHS is rewritten to achieve absolute convergence for <math>1<|z|\le2\pi</math>). [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 09:14, 25 December 2021 (UTC)
::::::::As I already explained, Riesel's and Gohl's (32) establishes an approximation, not an equality as <math>N\to\infty</math>. You were not able to provide a reliable source containing the proof of the equality. Original research is not allowed on Wikipedia, anyway. So, we can't have an equality instead of an approximation here. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 16:39, 25 December 2021 (UTC)
:You just blindly state what you want. OK, let's look at the final equality in the [http://www-m3.ma.tum.de/bornemann/RiemannRZero.pdf Bornemann's paper]:
:<math>\mathrm{R}(e^{-2\pi t})=\frac{1}{\pi}\sum\limits_{k=1}^{\infty}\frac{(-1)^{k-1}t^{-2k-1}}{(2k+1)\zeta(2k+1)}+\frac12\sum\limits_{\rho}\frac{t^{-\rho}}{\rho\cos(\pi\rho/2)\zeta'(\rho)}</math>
:and ''directly'' calculate the sum questioned:
:<math>\sum\limits_{m=1}^{\infty}\mathrm{R}(e^{-2\pi mt})=\frac{1}{\pi}\sum\limits_{k=1}^{\infty}\frac{(-1)^{k-1}t^{-2k-1}\zeta(2k+1)}{(2k+1)\zeta(2k+1)}+\frac12\sum\limits_{\rho}\frac{t^{-\rho}\zeta(\rho)}{\rho\cos(\pi\rho/2)\zeta'(\rho)}=\frac{1}{\pi}\left(\frac1t-\arctan\frac1t\right)</math>
:Now it is pretty obvious without any analytic number theory tricks.
:{{ping|A1E6}} Merry Christmas! [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 17:41, 25 December 2021 (UTC)
::The last result in Bornemann's paper is
::<math>R(e^{-2\pi t})=\frac{1}{\pi}\sum_{k=1}^\infty\frac{(-1)^{k-1}t^{-2k-1}}{(2k+1)\zeta(2k+1)}+\frac12\sum_{\rho}\frac{t^{-\rho}}{\rho\cos(\pi\rho/2)\zeta'(\rho)}</math>
::where <math>\rho</math> runs over the non-trivial zeros of the Riemann zeta function. Concluding that
::<math>\sum_{m=1}^{\infty}R(e^{-2\pi mt})=\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}t^{-2k-1}\zeta(2k+1)}{(2k+1)\zeta(2k+1)}+\frac12\sum_{\rho}\frac{t^{-\rho}\zeta(\rho)}{\rho\cos(\pi\rho/2)\zeta'(\rho)}=\frac{1}{\pi}\left(\frac1t-\arctan\frac1t\right)</math>
::is your original research and unless you provide a reliable source with a proof of this implication, we can't have that on Wikipedia. So I have to revert your edit again. Merry Christmas. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 18:03, 25 December 2021 (UTC)
:::Are you kidding me? You really need a proof than <math>\sum_{m=1}^{\infty}m^{-s}=\zeta(s)</math>? [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 22:22, 25 December 2021 (UTC)
::::That is the definition of the Riemann zeta function and does not prove your result by any means.   And the two sources you provided do ''not'' prove your result. You do your original research supposedly based on them. Anyone who understands Wikipedia policy will see that you are trying to sneak in your original research. Please stop it. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 22:36, 25 December 2021 (UTC)
:::::Anyone who understands analysis will see that the equality straightly follows from both sources (most easily from Bornemann), so that's not original research but a clear fact. I think you are lying when you say ''does not prove your result by any means'', because the proof is one-line and obvious. You are continuously trying to put wrong information on Wikipedia, that's just vandalism. I am highly disappointed. [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 06:11, 26 December 2021 (UTC)
::::::"''You are continuously trying to put wrong information on Wikipedia''" I'm not; I'm trying to keep misinformation off of Wikipedia and trying to enforce Wikipedia policies. It is confirmed that <math>L\approx R</math>, but <math>L=R</math> is a bolder statement and any proof that <math>L=R</math> has to be in a reliable source, otherwise <math>L=R</math> cannot be included in the article. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 10:18, 26 December 2021 (UTC)
:::::::You ''are'' since you insist that the equality does not hold while it really does. A very detailed proof is given in the article now, and the sources are clearly provided. Any reader who understand the rest of the article can easily understand that short proof, and the most of readers can see how the equality follows from Riesel & Göhl. If you can't, please ask those who understand the basics of analytical number theory. But please don't blindly revert anything, that looks ridiculous at least. I hope the question is resolved now. [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 16:41, 26 December 2021 (UTC)
::::::::Now you added your own proof to the article. The proof is considered original research, since Bornemann's paper itself does not prove your result, nor can your proof be found in any reliable source. "''ask those who understand the basics of analytical number theory''" It doesn't matter at all whether you understand analytic number theory, you're breaking Wikipedia policies by adding your original research. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 16:49, 26 December 2021 (UTC)
:::::::::It seems you are the only person who consider that obvious consequence as an original research. I have no more time to deal with this, sorry. [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 16:59, 26 December 2021 (UTC)
::::::::::"''You ''are'' since you insist that the equality does not hold''" I'm sorry for an improper choice of words in my first edit summary, namely "''The arctan-log terms are not equivalent to the sum over the trivial zeta zeros''" It is true that the equality was not disproved. The proper wording would be "''Riesel's and Gohl's (32) does not prove equality etc.''" In any case <math>L\approx R</math> does ''not'' contradict <math>L=R</math>, so you really don't have to be worried that it contradicts your result. But still, unless you provide a reliable source with a proof of the bold statement <math>L=R</math>, we can't have it in the article. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 17:16, 26 December 2021 (UTC)
::::::::::{{reply to|Droog Andrey}} I added Bornemann's result, but your proof still has to be in a reliable source. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 18:52, 26 December 2021 (UTC)
:::::::::::The proof based on Bornemann's paper was introduced mostly to convince ''you'' that the equality holds. Actually, the equality straightly follows from Riesel&Göhl. [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 22:21, 26 December 2021 (UTC)
::::::::::::Riesel and Gohl do not use zeta regularization. They do not interpret <math display="inline">\sum_{n=1}^\infty \mu (n)</math> as <math>-2</math>. All they do is compute the sum up to some natural number <math>N</math>. Then they choose <math>N=154</math>. And for <math>N=154</math> we have <math display="inline">\sum_{n=1}^{154}\mu (n)=-2</math>. The equality of yours does not follow from Riesel and Gohl. Regarding Bornemann, additional and significant research is needed to obtain your result from it. And that is exactly what you're trying to do, original research. Your work is ''very'' far from [[WP:CALC]]. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 23:08, 26 December 2021 (UTC)
:::::::::::::No it's not. The first step is just literally substituting in the definition of the zeta function. The second is remembering what the &rho; are. And the third is remembering the series for the arctangent. None of this seems like stuff that even needs any knowledge of analytic number theory to me, much less "significant research". [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 16:43, 27 December 2021 (UTC)
::::::::::::::{{reply to|Double sharp}} I don't think so. For example,
::::::::::::::<math>\sum_{m=1}^\infty\sum_{\rho}\frac{m^{-\rho}t^{-\rho}}{\rho\cos(\pi\rho/2)\zeta'(\rho)}=\sum_{\rho}\sum_{m=1}^\infty \frac{m^{-\rho}t^{-\rho}}{\rho\cos(\pi\rho/2)\zeta'(\rho)}=0</math>
::::::::::::::(which Droog Andrey used) is not [[WP:CALC]]. In particular, the order of infinite summation is interchanged, without justification. The interchange is allowed when
::::::::::::::<math>\sum_{m=1}^\infty\sum_{\rho}\left|\frac{m^{-\rho}t^{-\rho}}{\rho\cos(\pi\rho/2)\zeta'(\rho)}\right|<\infty</math>,
::::::::::::::but this was not proved: perhaps intentionally, to make the proof seem like a one-liner and not original research. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 19:16, 27 December 2021 (UTC)
::::::::::::::Also, importantly, Bornemann used a simplification which depends on the assumption that all zeros of the Riemann zeta function are simple (footnote on page 1), but that is a mere conjecture. And see "''The poles of <math>f_\tau</math> are all simple and they are located at the poles of <math>\Gamma (-z)</math>, i.e., at <math>z = 0, 1, 2, \ldots</math> and at the values <math>z</math> for which <math>\zeta (z+1)=0</math>.''" (page 2). [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 23:25, 27 December 2021 (UTC)
:{{reply to|A1E6}} Riesel&Göhl did use <math display="inline">\sum_{n=1}^\infty \mu(n) = -2</math>, that's why they chose <math>N=154</math>. Summation interchange is obviously legal, just look at the denominator. Zeros simplicity doesn't matter at all (see the footnote on the 1st page of Bornemann). BTW you continuously mess up references to Riesel&Göhl. That doesn't seem like you care about the quality of the article. [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 08:20, 28 December 2021 (UTC)
::They did ''not'' use <math display="inline">\sum_{n=1}^\infty \mu (n)=-2</math> because <math display="inline">\sum_{n=1}^\infty \mu (n)=-2</math> is literally nowhere in their article, and <math>154\ne\infty</math>. The summation interchange
::<math>\sum_{m=1}^\infty\sum_{\rho}\frac{m^{-\rho}t^{-\rho}}{\rho\cos(\pi\rho/2)\zeta'(\rho)}=\sum_{\rho}\sum_{m=1}^\infty \frac{m^{-\rho}t^{-\rho}}{\rho\cos(\pi\rho/2)\zeta'(\rho)}</math>
::is not obviously legal and was not proved by you. Even if you could provide a proof that it is legal, it would still count as original research. Zeros simplicity ''does'' matter, please read carefully the text I quoted from page 2. Also, you misinterpreted the footnote on page 1. Assuming the conjecture is not essential for the goal of his paper, namely for finding the largest positive zero of Riemann's <math>\operatorname{R}</math>. It is, however, essential for simplifying the terms of the infinite series running over all zeros. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 11:58, 28 December 2021 (UTC)
::By the way, you can ignore "''The arctan-log terms *are not* equivalent to the integral term''" in Vitamindeth's edit summary from 30 July 2021. The proper wording would be "''It was not proved that the arctan-log terms are equivalent to the integral term''". And even if <math>L=R</math>, stating that <math>L\approx R</math> would not be a "flaw". [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 12:33, 28 December 2021 (UTC)
:::<math display="inline">\sum_{n=1}^\infty \mu (n)=-2</math> was the reason to choose such <math>N</math> that <math display="inline">\sum_{n=1}^N \mu (n)=-2</math>. On summation interchange: please don't forget that non-trivial zeros are ordered by their height. Terms simplifying is not essential because <math>\sum_m = 0</math> for every non-trivial zeta zero anyway. It seems that you ask for every single step just because you don't want to recognize the proof to be actually simple enough to fit [[WP:CALC]] in the article's context. [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 21:13, 28 December 2021 (UTC)
::::"''<math display="inline">\sum_{n=1}^\infty \mu (n)=-2</math> was the reason to choose such <math>N</math> that <math display="inline">\sum_{n=1}^N \mu (n)=-2</math>''" This is false.

::::Let me quote p. 979 from Riesel and Gohl: "''It is thus advantageous to choose such a value of <math>N</math> that the sums <math>g_k</math> become comparatively small. It is also advantageous to have <math display="inline>\sum_{n=1}^N \mu (n)=-2</math> at the same time, '''since (32) then has the order of magnitude only <math>=O\left((\log  x)^{-3}\right)</math> instead of <math>O\left((\log x)^{-1}\right)</math>.''' This turns out to be the case for (among some other small values of <math>N</math>) <math>N=154</math>. In our computation we have chosen this value of <math>N</math> in (14) and (19).''"

:::::That's actually the same reason since the order of magnitude damping comes from <math display="inline>\sum_{n=1}^{\infty} \mu (n)=-2</math>. [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 00:35, 29 December 2021 (UTC)

::::You state "''On summation interchange: please don't forget that non-trivial zeros are ordered by their height.''" This does not prove
::::<math>\sum_{m=1}^\infty\sum_{\rho}\left|\frac{m^{-\rho}t^{-\rho}}{\rho\cos(\pi\rho/2)\zeta'(\rho)}\right|<\infty .</math>

:::::Absolute convergence is not needed there because of zeta zeros ordering. [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 00:35, 29 December 2021 (UTC)

::::Then you state "''Terms simplifying is not essential because <math>\sum_m = 0</math> for every non-trivial zeta zero anyway.''" To me, it seems like you didn't even bother to read the quoted text from Bornemann's page 2. Bornemann is not summing over <math>m</math> to obtain <math>\zeta (\rho)</math> in the numerator, only you are. Bornemann's result depends on assuming the conjecture, as I've already explained several times. And you're trying to derive your result from Bornemann's. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 22:44, 28 December 2021 (UTC)

:::::I've read a whole article a while ago. The possible changes in Bornemann's result due to zeta zeros multiplicity do not affect the sum over <math>m</math>.
:::::It seems that repeating the same thing in different words does not help, huh. Let's stop for a while and see what others have to say. [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 00:35, 29 December 2021 (UTC)
::::::When you start your proof with
::::::<math>\operatorname{R}(e^{-2\pi t})=\frac{1}{\pi}\sum_{k=1}^\infty\frac{(-1)^{k-1}t^{-2k-1}}{(2k+1)\zeta(2k+1)}+\frac12\sum_{\rho}\frac{t^{-\rho}}{\rho\cos(\pi\rho/2)\zeta'(\rho)}</math>
::::::(you did), you use Bornemann's fully-fledged result depending on the conjecture, for the reasons explained above. Not to mention the fact that the first result of Bornemann's paper is
::::::<math>\operatorname{R}(e^{-\tau})=\sum_{\rho}\frac{\Gamma (1-\rho)}{(\rho -1)\zeta '(\rho)}\tau ^{\rho -1}</math>
::::::where <math>\rho</math> runs over all zeros of the Riemann zeta function and <math>\tau >0</math> – and ''even this'' was proved by Bornemann by assuming the conjecture that all zeros of the Riemann zeta function are simple. In particular, on page 2, he used
::::::<math>\operatorname{Res}_{z=\rho -1}\left(-\frac{\Gamma (-z)\tau^z}{z\zeta (z+1)}\right)=\frac{\Gamma (1-\rho)\tau ^{\rho -1}}{(1-\rho)\zeta '(\rho)},</math>
::::::exploiting the simplicity of the poles due to the conjectured simplicity of the zeros of the Riemann zeta function. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 01:21, 29 December 2021 (UTC)
:::::::That's nice, but <math>\tau^{1-\rho}</math> will not disappear from the denominator once you encounter a multiple zero. Please keep track of essential details instead of finding faults everywhere. [[User:Droog Andrey|Droog Andrey]] ([[User talk:Droog Andrey|talk]]) 07:54, 29 December 2021 (UTC)
::::::::"''finding faults everywhere''" – ''that'' is essential here. All zeros of <math>\zeta (s)</math> on <math>\Re s=\tfrac12</math> are simple if and only if <math>\zeta '(s)\ne 0</math> on <math>\Re s=\tfrac12</math>. And it is not even known whether there exists a <math>\rho</math> such that <math>\zeta (\rho)=\zeta '(\rho)=0</math>. [[User:A1E6|A1E6]] ([[User talk:A1E6|talk]]) 17:49, 3 January 2022 (UTC)
{{reflist-talk}}

== Agama result ==

[[User:Sadinova|Sadinova]] ([[Special:Contributions/Sadinova]]) has twice added (in January and today) a paper of Agama to the article purporting to give a new formula for pi(x) in terms of Chebyshev's function vartheta(x). Basic asymptotic analysis shows that the formula varies as x/(2 log x) and so can't be correct. (I also computed it at powers of 10 to check.) But even if correct, this isn't a useful formula since Chebyshev's formula is hard to compute. If there was some other reason to care about this incorrect result I wouldn't be opposed but I don't see it.

[[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 16:27, 23 March 2023 (UTC)

:Dear Great house, the formula does not vary as x/2logx. Rather it varies as x/2log x minus the term contributed by the sum. The utility can be challenged as does the utility of the one existing in the literature. Sometimes these things becomes usuful in theory and not for computational persuits. [[User:Sadinova|Sadinova]] ([[User talk:Sadinova|talk]]) 20:38, 26 March 2023 (UTC)
:I propose a replacement of the result, because it is more explicit than the one we currently have on the page. It might be useful in theory. Nevertheless, of gives a good estimate for the integrals involving the prime counting function and Chebyshev's function. [[User:Sadinova|Sadinova]] ([[User talk:Sadinova|talk]]) 20:47, 26 March 2023 (UTC)
::Having seen some other works by Agama, such as [https://arxiv.org/abs/1706.09803 his supposed proof] of [[Goldbach's conjecture]], I would require independent verification of any claims by him before accepting them as a reliable source.
::As for Sadinova, as their only contributions have been to attempt to push work by Agama into Wikipedia, I would remind them that Wikipedia's terms of use demand that any [[WP:COI|conflict of interest]] be explicitly declared. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 23:20, 26 March 2023 (UTC)

::: Sadinova, if you learn of a correct version of this function, I would be happy to hear from you as I am curious. But I don't think it belongs in the article. Regardless, this incorrect version does not belong in this article.
::: As for the expression,
:::: <math>\frac{1}{\log x}\sum_{n\le x}\log(x/n) = \frac{x}{\log x} + O(1)</math>
::: by Stirling's approximation. The whole expression is hence
:::: <math>\frac{\vartheta(x)}{ \log x } +\frac{x}{2 \log x }-\frac{1}{ 4}- \frac{1}{\log x}\sum_{n\le x}\log (\frac{x}{n}) = \frac{\vartheta(x)}{\log x} - \frac{x}{2\log x} + O(1)</math>
::: which is
:::: <math>\frac{x}{2\log x} + O(x^{1/2+\varepsilon})</math>
::: under the Riemann hypothesis. (Without, the error term is larger, but the leading term is unchanged.) You can also compute the terms explicitly and see that they are about half of the function they purport to approximate.
::: [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 13:00, 29 March 2023 (UTC)
::::Dear Greathouse,
::::The sum runs over only odd composites and not the entire integers. You must have misconstrued the result. Kindly note that I've worked this out and checked thoroughly. Well, it might not be suitable here but it must be correct.
::::([[User:Sadinova|Sadinova]] ([[User talk:Sadinova|talk]]) 09:53, 4 April 2023 (UTC))

::::: Unlike the formula you added to the article, this one looks plausible. It's still not appropriate for the article, but I do find it interesting.
::::::: <math>\pi(x) \mathbin{\stackrel?=} \frac{\vartheta(x)}{ \log x } +\frac{x}{2 \log x }-\frac{1}{ 4}- \frac{1}{\log x}\!\!\!\!\!\!\!\!\!\sum_{\stackrel{n\le x}{n\text{ odd composite}}}\log \left(\frac{x}{n}\right)+O\left(\frac{1}{\log x}\right)</math>
::::: In fairness, the numerics don't support an error term quite that sharp, it looks more like O(log log x) or O(1). But I'd have to dive deeper in this to verify.
::::: [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 20:28, 5 April 2023 (UTC)

== Error in a 2018 paper ==

I was having a look at {{cite journal |title=Improved Explicit Bounds for Some Functions of Prime Numbers |first=Sadegh |last=Nazardonyavi |journal=Functiones et Approximatio |volume=58 |issue=1 |date=March 2018 |pages=7–22 |doi=10.7169/facm/1547 |url=https://projecteuclid.org/journals/functiones-et-approximatio-commentarii-mathematici/volume-58/issue-1/Improved-explicit-bounds-for-some-functions-of-prime-numbers/10.7169/facm/1547.pdf}} and was trying to sharpen the bound on Proposition 3.3 (p. 14), which claims:
:<math>\pi(x) < \frac{x}{\log x}\left(1 + \frac{1.0794}{\log x}\right),\quad\text{for }x \ge 8\cdot 10^{11}.</math>
When I test that with <math>x = 800,000,000,000</math>, I get
:<math>\pi(x) = 30,341,383,527</math>[https://sweet.ua.pt/tos/primes.html]
:<math>\log x = 27.407\,877\,564\,614\,338\,452\,449\,602\,365\,903</math>
:<math>\frac{x}{\log x}\left(1 +\frac{1.0794}{\log x}\right) = 30,338,221,862.669 < \pi(x)</math>

Which is noticeably ''less'' than the value it's supposed to be an upper bound for.  Can someone check my arithmetic?

{{abbr|FWIW|For what it's worth}}, I tested the paper's other upper bound
:<math>\pi(x) < \frac{x}{\log x}\left(1 + \frac{1}{\log x} + \frac{2.256}{\log^2 x}\right)</math>
for <math>x < 8\cdot 10^{11},</math> and it holds for <math>x > 406,742,886,707.225</math>, where <math>\pi(x)=15,833,338,468</math> for <math>406,742,886,673 \leq x < 406,742,886,757.</math> [[Special:Contributions/97.102.205.224|97.102.205.224]] ([[User talk:97.102.205.224|talk]]) 18:34, 6 November 2024 (UTC)

: I don't really follow the math, but I figured out the derivation of 1.0794 in the proof of Proposition 3.3.  It comes from setting <math>A_1^'(x_0) = A_2^'(x_0)</math> and solving for {{mvar|c}}.
: First, I have to correct an error in the derivative of <math>A_1(x)</math>.  I limited myself to the case of {{math|1=''k'' = 1}} to simplify my life, and in that case, the derivative of {{math|''A''<sub>2</sub>(''x'')}} is given correctly in the paper:
:: <math>A_2^'(x) = \frac{1}{\log x} + \frac{c-1}{\log^2 x} - \frac{2c}{\log^3 x}.</math>
: The derivative given for {{math|''A''<sub>1</sub>(''x'')}}, however, is All Wrong.  There shouldn't be any {{mvar|x}}s in the numerator!
: {{math|''A''<sub>1</sub>}} has two terms.  The first (non-integral) term looks just like {{math|''A''<sub>2</sub>}}, with {{mvar|&eta;}} instead of {{mvar|c}}, and its derivative is also the same (making the same substitution).  (I'm using plain {{mvar|&eta;}}, omitting the subscript {{mvar|k}} and prime symbol, to mimimize my typing.)
: The derivative of the second (integral) term of {{math|''A''<sub>1</sub>}} is simply the integrand: <math>\frac{1}{\log^2 x}\left(1 + \frac{\eta}{\log x}\right) = \frac{1}{\log^2 x} + \frac{\eta}{\log^3 x}.</math>
: Adding these two parts together, we have
:: <math>\begin{align}
A_1^'(x) &= \frac{1}{\log x} + \frac{\eta-1}{\log^2 x} - \frac{2\eta}{\log^3 x} + \frac{1}{\log^2 x} + \frac{\eta} {\log^3 x} \\
         &= \frac{1}{\log x} + \frac{\eta}{\log^2 x} - \frac{\eta}{\log^3 x}
\end{align}</math>
: Setting the two parts equal, we get
::<math>\frac{1}{\log x} + \frac{\eta}{\log^2 x} - \frac{\eta}{\log^3 x} = \frac{1}{\log x} + \frac{c-1}{\log^2 x} - \frac{2c}{\log^3 x}.</math>
: Cancelling the common <math>\frac{1}{\log x}</math> and multiplying the rest by <math>\log^3 x</math> (which is positive in the cases we care about), this produces
::<math>\eta\log x - \eta = (c-1)\log x - 2c.</math>
: Solving for {{mvar|c}}, we get
::<math>\eta\log x + \log x - \eta = c\log x - 2c,</math>
::<math>\eta\log x + \log x - \eta = c(\log x - 2),</math>
::<math>\frac{\eta\log x + \log x - \eta}{\log x - 2} = c.</math>
: Given the value {{math|1=''&eta;'' = 0.000648713234}} for {{math|1=''k'' = 1}} from the table in Proposition 3.1, the left-hand side comes out as a constant plus a negative number divided by {{math|log ''x'' &minus; 2}}, so it's a decreasing function and the lower bound is set by the minimum value of {{mvar|x}}.  Substituting {{math|1=log ''x'' = log 8⋅10<sup>11</sup> = 27.407877&hellip;}}, this produces
::<math>c = 1.0793899897592</math>
: which the paper has rounded up to 1.0794.  So that part I understand.  But the source of the {{mvar|&eta;}} value is very mysterious, so a typo there could have messed everything up.
: But there's a second inequality to check.  The goal is to prove that {{math|''A''<sub>1</sub>(''x'')  < ''A''<sub>2</sub>(''x'')}} for all {{math|1=''x'' ≥ ''x''<sub>0</sub> = 8⋅10<sup>11</sup>}}, which is done by ensuring {{math|''A''<sub>1</sub>(''x''<sub>0</sub>) < ''A''<sub>2</sub>(''x''<sub>0</sub>)}} ''and'' the derivatives compare correctly for {{math|''x'' ≥ ''x''<sub>0</sub>}}.  We've checked the latter, but not the former.
: Subtracting <math>\frac{x}{\log x}</math> from both {{math|''A''<sub>1</sub>}} and {{math|''A''<sub>2</sub>}}, and asking a symbolic math package for help with the integram, we get a left hand side of {{math|1153448872.0718445163196688097324}} and a right hand side of {{math|1064974418.62797''c''}}.  To preserve our inequality, we require {{math|''c'' > 1.0830765996782}}. '''Bingo!'''  I bet this was the author's mistake.
: Plugging that value into the original equation results in an upper bound that actually exceeds {{math|''π''(''x''<sub>0</sub>)}} [[Special:Contributions/97.102.205.224|97.102.205.224]] ([[User talk:97.102.205.224|talk]]) 02:56, 7 November 2024 (UTC)

== Prime index function ==

Epstein, consider incorporating the recent edit concerning the “prime index function “ into the main article. If you have any strong reason for this to be excluded, it has to be done with reason. You cannot just remove contents on a whim. This course of action should be demonstrated with good reason. If you do not do this, I will revert the previous edit. Try to uphold some respect and professionalism. [[Special:Contributions/132.203.184.80|132.203.184.80]] ([[User talk:132.203.184.80|talk]]) 22:42, 23 February 2025 (UTC)

:We do not generally include published results merely because they are published results. There needs to be some evidence that other researchers consider them important to the topic. There is no such evidence here. We have only a 2020 research paper, summarized but not even reviewed by [[MathSciNet]] and [[zbMATH]], and cited by absolutely nobody else.
:To put it another way: Google Scholar gives some 2600 papers that use the phrase "prime counting function". We are only citing 40 of those, about 1.5%. What gives you the idea that this paper is in the top 1.5% of the most significant works about the prime counting function, let alone so important that it deserves its own section in the article? —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 23:20, 23 February 2025 (UTC)
::Hi Epstein, the idea that this topic has garnered little attention is no good reason to assume it is not worthy of inclusion as public knowledge. If these are standards of Wikipedia, then we are probably getting it all wrong with the rules. I don’t know if you have read the research paper. The potential utility cannot be overemphasized. [[Special:Contributions/132.203.184.80|132.203.184.80]] ([[User talk:132.203.184.80|talk]]) 00:29, 24 February 2025 (UTC)
:::Looking more carefully I notice that the author of the work in question is also responsible for [[viXra]] preprints claiming solutions of the P=NP problem and "an elementary proof of Fermat's Last Theorem". Also the work does not appear to have been carefully copyedited; it spells Riemann wrong in the second line. The first of the two theorems claimed in the paper's introduction is false (it claims that for real intervals on which the parity of the number of primes is equal at the start and end of the interval, there exists a prime in the interval, false for any interval that does not contain a prime number). The second theorem claimed in the introduction is totally trivial (each non-prime number is the center of an interval within which the parity of the number of primes does not change). The need to take everything by this author with a boatload of salt cannot be overemphasized. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 00:57, 24 February 2025 (UTC)
::::You should learn to humble yourself and know that no one is a silo of knowledge. Every scribble in that paper is well thought of and is correct. You cannot read that paper with the lens you seem to have and make a tangible meaning out of it. You cannot challenge  the content of that paper because you seem to exude a problematic mindset, perhaps a conventional one akin to most folks in the community. Forgive me for my characterization but you typically reflect this way of thinking. You can write to the editor of the journal purporting that the paper is wrong with your justification. When you mentioned vixra as some outlet of the author, what do you mean to imply? [[Special:Contributions/132.203.184.80|132.203.184.80]] ([[User talk:132.203.184.80|talk]]) 02:00, 24 February 2025 (UTC)
:::::By the way, your IP address geolocates to U. Laval. Theophilus Agama, the author of this work, is listed online as being a student at U. Laval. If perchance you have some connection to or [[WP:COI|conflict of interest]] involving Agama, it is a violation of Wikipedia's terms of use to fail to disclose it. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 02:39, 24 February 2025 (UTC)
::::::If perchance I read some of the authors work in an institutional repository without meeting in person, why should that violate Wikipedia policy. I think it is ideally prudent for us to refrain from this discussion. [[Special:Contributions/132.203.184.80|132.203.184.80]] ([[User talk:132.203.184.80|talk]]) 03:14, 24 February 2025 (UTC)

::::::: I support David on this. [[User:Bubba73|Bubba73]] <sup>[[User talk:Bubba73|You talkin' to me?]]</sup> 05:36, 24 February 2025 (UTC)
:::::::That's an oddly specific non-response to my implied question, so it led me to check. Laval does indeed have an institutional repository, https://corpus.ulaval.ca/, but there are no works by Agama in it, so your hypothetical explanation appears to be impossible. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 06:18, 24 February 2025 (UTC)
::::::::I tried to raise your consciousness regarding taking issues and matters for granted a short while ago. By the way, I never mentioned that I read the work of the author in the repository you referred to. Over assumptions and potential unconscious biases has to be guard against in your voluntary work. You are a professional and being mindful of this pitfall is as much important as your role. [[Special:Contributions/132.203.184.80|132.203.184.80]] ([[User talk:132.203.184.80|talk]]) 09:32, 24 February 2025 (UTC)
::::::::: The fact is that this paper was not reviewed by the AMS Reviews nor by the Zbl, which both chose to reproduce its summary only, and that the AMS Reviews lists 0 citation from References and 0 citation from Reviews, despite its publication 5 years ago. Its notability amongst number theorists is inexistent, and it thus should not be featured in a Wikipedia article. Wikipedia is not a showcase for subjects without notoriety.     --[[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 10:11, 24 February 2025 (UTC)
::::::::::Fair enough [[Special:Contributions/132.203.184.80|132.203.184.80]] ([[User talk:132.203.184.80|talk]]) 11:31, 24 February 2025 (UTC)
:::: After that discovery, David, I'm quite sure I don't want to see the article cited here without extraordinary circumstances. Good find. - [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 19:27, 25 February 2025 (UTC)
:::::Well, your affirmation is possibly grounded on a misplaced judgement. This is a knowledge-based forum and one has to careful about how issues of this nature ought to handled, upholding a sense of objectivity without allowing bias to cloud your judgment. There are loopholes in your approaches and you need to acknowledge this shortfall. If you find comfort in dismissing contributions that do not align with your thought process, then am sorry for you. That’s certainly not the way to do and popularize science. [[Special:Contributions/132.203.184.80|132.203.184.80]] ([[User talk:132.203.184.80|talk]]) 15:15, 2 March 2025 (UTC)
:::::: I'm not going to support Wikipedia citing cranks or [[Pseudomathematics|pseudomathematicians]]. The paper was already out for notability reasons, but this gives strong reason to doubt its veracity which strikes to the heart of why we would cite it in the first place. - [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 19:48, 3 March 2025 (UTC)
:::::::You are not under any compulsion to endorse any editing attempt from a ''Wiki citing crank'' or a ''pseudo mathematician''. If you feel at ease with your ego-driven mindset, it does not rule out the possibility of making grave errors. Until you learn to remain humbled and accepting that we may not always have all the answers, and that open discussion and reception to diverse view points is key to enlightenment and understanding, you may not be headed anywhere. [[Special:Contributions/132.203.211.210|132.203.211.210]] ([[User talk:132.203.211.210|talk]]) 21:08, 3 March 2025 (UTC)
:::::::: What use is it to keep this pointless rambling discussion going on and on? Your attempted contribution was inadequate, period. This is not a forum liable to heal hurt feelings.--[[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 23:28, 3 March 2025 (UTC)