View source for Mertens' theorems

{{short description|Three results related to the density of prime numbers}}
{{For|Mertens's theorem on convergence of Cauchy products of series|Cauchy product#Convergence and Mertens' theorem}}
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In [[analytic number theory]], '''Mertens' theorems''' are three 1874 results related to the density of [[prime number]]s proved by [[Franz Mertens]].<ref name="Mertens">F. Mertens. J. reine angew. Math. 78 (1874), 46–62 [http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0078&DMDID=dmdlog5 Ein Beitrag zur analytischen Zahlentheorie]</ref>

In the following, let <math>p\le n</math> mean all primes not exceeding ''n''.

== First theorem ==

'''Mertens' first theorem''' is that

:<math> \sum_{p \le n} \frac{\log p}{p} - \log n</math>

does not exceed 2 in absolute value for any <math> n\ge 2</math>. ({{OEIS link|id=A083343}})

== Second theorem ==

'''Mertens' second theorem''' is

:<math>\lim_{n\to\infty}\left(\sum_{p\le n}\frac1p -\log\log n-M\right) =0,</math>

where ''M'' is the [[Meissel–Mertens constant]] ({{OEIS link|id=A077761}}). More precisely, Mertens<ref name="Mertens"/> proves that the expression under the limit does not in absolute value exceed

:<math> \frac 4{\log(n+1)} +\frac 2{n\log n}</math>

for any <math> n\ge 2</math>.

=== Proof ===

The main step in the proof of Mertens' second theorem is 
:<math>O(n)+n\log n=\log n! =\sum_{p^k\le n} \lfloor n/p^k\rfloor\log p =
\sum_{p^k\le n} \left(\frac{n}{p^k}+O(1)\right)\log p= n \sum_{p^k\le n}\frac{\log p}{p^k}\ + O(n)</math>
where the last equality needs <math>\sum_{p^k\le n}\log p =O(n)</math> which follows from <math>\sum_{p\in (n,2n]}\log p\le \log{2n\choose n}=O(n)</math>.

Thus, we have proved that
:<math>\sum_{p^k\le n}\frac{\log p}{p^k}=\log n+O(1)</math>.
Since the sum over prime powers with <math>k \ge 2</math> converges, this implies
:<math>\sum_{p\le n}\frac{\log p}{p}=\log n+O(1)</math>.
A [[Summation by parts|partial summation]] yields
:<math>\sum_{p\le n} \frac1{p} = \log\log n+M+O(1/\log n)</math>.

=== Changes in sign ===

In a paper <ref>{{Cite journal |last=Robin |first=G. |year=1983 |title=Sur l'ordre maximum de la fonction somme des diviseurs |journal=Séminaire Delange–Pisot–Poitou, Théorie des nombres (1981–1982). Progress in Mathematics|volume=38 |pages=233–244 }}</ref> on the growth rate of the [[Divisor function#Approximate growth rate|sum-of-divisors function]] published in 1983, Guy Robin proved that in Mertens' 2nd theorem the difference

:<math>\sum_{p\le n}\frac1p -\log\log n-M</math>

changes sign infinitely often, and that in Mertens' 3rd theorem the difference

:<math>\log n\prod_{p\le n}\left(1-\frac1p\right)-e^{-\gamma}</math>

changes sign infinitely often. Robin's results are analogous to [[John Edensor Littlewood|Littlewood]]'s [[Skewes's number#Skewes's numbers|famous theorem]] that the difference π(''x'')&nbsp;−&nbsp;li(''x'') changes sign infinitely often. No analog of the [[Skewes number]] (an upper bound on the first [[natural number]] ''x'' for which π(''x'')&nbsp;>&nbsp;li(''x'')) is known in the case of Mertens' 2nd and 3rd theorems.

=== Relation to the prime number theorem <span class="anchor" id="Mertens' second theorem and the prime number theorem"></span> ===
Regarding this asymptotic formula Mertens refers in his paper to "two curious formula of Legendre",<ref name="Mertens"/> the first one being Mertens' second theorem's prototype (and the second one being Mertens' third theorem's prototype: see the very first lines of the paper). He recalls that it is contained in Legendre's third edition of his "Théorie des nombres" (1830; it is in fact already mentioned in the second edition, 1808), and also that a more elaborate version was proved by [[Pafnuty Chebyshev|Chebyshev]] in 1851.<ref>P.L. Tchebychev. Sur la fonction qui détermine la totalité des nombres premiers. Mémoires présentés à l'Académie Impériale des Sciences de St-Pétersbourg par divers savants, VI 1851, 141–157</ref> Note that, already in 1737, [[Leonhard Euler|Euler]] knew the asymptotic behaviour of this sum.

Mertens diplomatically describes his proof as more precise and rigorous. In reality none of the previous proofs are acceptable by modern standards: Euler's computations involve the infinity (and the hyperbolic [[logarithm]] of infinity, and the logarithm of the logarithm of infinity!); Legendre's argument is [[heuristic]]; and Chebyshev's proof, although perfectly sound, makes use of the Legendre-Gauss conjecture, which was not proved until 1896 and became better known as the [[prime number theorem]].

Mertens' proof does not appeal to any unproved hypothesis (in 1874), and only to elementary [[real analysis]]. It comes 22 years before the first proof of the prime number theorem which, by contrast, relies on a careful analysis of the behavior of the [[Riemann zeta function]] as a function of a complex variable.
Mertens' proof is in that respect remarkable. Indeed, with [[Big O notation|modern notation]] it yields
:<math>\sum_{p\le x}\frac1p=\log\log x+M+O(1/\log x)</math>
whereas the prime number theorem (in its simplest form, without error estimate), can be shown to imply <ref> I.3 of: G. Tenenbaum. Introduction to analytic and probabilistic number theory. Translated from the second French edition (1995) by C. B. Thomas. Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge,1995.</ref>
:<math>\sum_{p\le x}\frac1p=\log\log x+M+o(1/\log x).</math>
In 1909 [[Edmund Landau]], by using the best version of the prime number theorem then at his disposition, proved<ref>Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909, Repr.  Chelsea New York 1953, § 55, p. 197-203.</ref> that 
:<math>\sum_{p\le x}\frac1p=\log\log x+M+O(e^{-(\log x)^{1/14}})</math>
holds; in particular the error term is smaller than <math>1/(\log x)^k</math> for any fixed integer ''k''. A simple [[summation by parts]] exploiting the [[Prime number theorem#Prime-counting function in terms of the logarithmic integral|strongest form known]] of the prime number theorem improves this to
:<math>\sum_{p\le x}\frac1p=\log\log x+M+O(e^{-c(\log x)^{3/5}(\log\log x)^{-1/5}})</math>
for some <math>c > 0</math>.

Similarly a partial summation shows that <math>\sum_{p\le x} \frac{\log p}{p} = \log x+ C+o(1)</math> is implied by the PNT.

== Third theorem ==

'''Mertens' third theorem''' is

:<math>\lim_{n\to\infty}\log n\prod_{p\le n}\left(1-\frac1p\right)=e^{-\gamma} \approx 0.561459483566885,</math>

where γ is the [[Euler–Mascheroni constant]] ({{OEIS link|id=A001620}}).

=== Relation to sieve theory<span class="anchor" id="Mertens' third theorem and sieve theory"></span> ===

An estimate of the probability of <math>X</math> (<math>X \gg n</math>) having no factor <math>\le n</math> is given by

:<math>\prod_{p\le n}\left(1-\frac1p\right)</math>

This is closely related to Mertens' third theorem which gives an asymptotic approximation of

:<math>P(p \nmid X\ \forall p \le n) = \frac{1}{e^\gamma \log n }</math>

==References==
{{Reflist}}

==Further reading==
*[[Akiva Moiseevich Yaglom|Yaglom]] and [[Isaak Moiseevich Yaglom|Yaglom]] ''Challenging mathematical problems with elementary solutions'' Vol 2, problems 171, 173, 174

==External links==
*{{MathWorld|urlname=MertensConstant|title=Mertens Constant}}


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[[Category:Summability theory]]
[[Category:Theorems about prime numbers]]