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The exact formula: spacing \, instead of [5px] which shows as "[5px]" in CHTML MathJax rendering

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{{Short description|Mathematical function}}
{{log(x)}}
[[Image:ChebyshevPsi.png|thumb|right|The Chebyshev function <math>\psi(x)</math>, with {{math|''x'' < 50}}]]
[[Image:Chebyshev.svg|thumb|right|The function <math>\psi(x)-x</math>, for {{math|''x'' < 104}}]]
[[Image:Chebyshev-big.svg|thumb|right|The function <math>\psi(x)-x</math>, for {{math|''x'' < 107}}]]

In [[mathematics]], the '''Chebyshev function''' is either a scalarising function ('''Tchebycheff function''') or one of two related functions. The '''first Chebyshev function''' {{math|''ϑ''&hairsp;&hairsp;(''x'')}} or {{math|''θ''&hairsp;(''x'')}} is given by

:<math>\vartheta(x) = \sum_{p \le x} \log p</math>

where <math>\log</math> denotes the [[natural logarithm]], with the sum extending over all [[prime number]]s {{mvar|p}} that are less than or equal to {{mvar|x}}.

The '''second Chebyshev function''' {{math|''ψ''&hairsp;(''x'')}} is defined similarly, with the sum extending over all [[prime power]]s not exceeding {{mvar|x}}

:<math>\psi(x) = \sum_{k \in \mathbb{N}}\sum_{p^k \le x}\log p = \sum_{n \leq x} \Lambda(n) = \sum_{p \le x}\left\lfloor\log_p x\right\rfloor\log p,</math>

where {{math|Λ}} is the [[von Mangoldt function]]. The Chebyshev functions, especially the second one {{math|''ψ''&hairsp;(''x'')}}, are often used in [[mathematical proof|proofs]] related to [[prime number]]s, because it is typically simpler to work with them than with the [[prime-counting function]], {{math|''π''&hairsp;(''x'')}} (see [[#The exact formula|the exact formula]] below.) Both Chebyshev functions are asymptotic to {{mvar|x}}, a statement equivalent to the [[prime number theorem]].

'''Tchebycheff function''', '''Chebyshev utility function''', or '''weighted Tchebycheff scalarizing function''' is used when one has several functions to be minimized and one wants to "scalarize" them to a single function:

:<math>f_{Tchb}(x,w) = \max_i w_i f_i(x).</math><ref name=JK>{{cite web|url=http://syllabus.cs.manchester.ac.uk/pgt/2017/COMP60342/COMP60342-2014-MOO.pdf|title=Multiobjective Optimization Concepts, Algorithms and Performance Measures|author=Joshua Knowles|date=2 May 2014|publisher=The University of Manchester|page=34}}</ref>

By minimizing this function for different values of <math>w</math>, one obtains every point on a [[Pareto front]], even in the nonconvex parts.<ref name=JK/> Often the functions to be minimized are not <math>f_i</math> but <math>|f_i-z_i^*|</math> for some scalars <math>z_i^*</math>. Then <math>f_{Tchb}(x,w) = \max_i w_i |f_i(x)-z_i^*|.</math><ref>{{cite journal|url=https://pure.tudelft.nl/ws/portalfiles/portal/30882193/FinalRevised_Improved_MOEA_D_for_BOPs_with_complicated_PFs.pdf|at=Page 6 equation (2)|title=An improved MOEA/D algorithm for bi-objective optimization problems with complex Pareto fronts and its application to structural optimization|last1=Ho-Huu |first1=V. |last2=Hartjes |first2=S. |last3=Visser |first3=H. G. |last4=Curran |first4=R. |journal=Expert Systems with Applications|doi=10.1016/j.eswa.2017.09.051|publisher=Delft University of Technology|date=2018}}</ref>

All three functions are named in honour of [[Pafnuty Chebyshev]].

==Relationships==
The second Chebyshev function can be seen to be related to the first by writing it as

:<math>\psi(x) = \sum_{p \le x}k \log p</math>

where {{mvar|k}} is the unique [[integer]] such that {{math|''p''&hairsp;''k'' ≤ ''x''}} and {{math|''x'' < ''p''&hairsp;''k'' +&hairsp;1}}. The values of {{mvar|k}} are given in {{OEIS2C|id=A206722}}. A more direct relationship is given by

:<math>\psi(x) = \sum_{n=1}^\infty \vartheta\big(x^{\frac{1}{n}}\big).</math>

This last sum has only a finite number of non-vanishing terms, as

:<math>\vartheta\big(x^{\frac{1}{n}}\big) = 0\quad \text{for}\quad n>\log_2 x = \frac{\log x}{\log 2}.</math>

The second Chebyshev function is the logarithm of the [[least common multiple]] of the integers from 1 to {{mvar|n}}.

:<math>\operatorname{lcm}(1,2,\dots,n) = e^{\psi(n)}.</math>

Values of {{math|lcm(1, 2, ..., ''n'')}} for the integer variable {{mvar|n}} are given at {{OEIS2C|id=A003418}}.

== Relationships between ''ψ''(''x'')/''x'' and ''ϑ''(''x'')/''x'' ==
The following [[theorem]] relates the two quotients <math>\frac{\psi(x)}{x}</math> and <math>\frac{\vartheta(x)}{x}</math> .<ref>{{Cite book |last=Apostol |first=Tom M. |title=Introduction to Analytic Number Theory |publisher=Springer |year=2010 |pages=75–76}}</ref>

'''Theorem:''' For <math>x>0</math>, we have

:<math>0 \leq \frac{\psi(x)}{x}-\frac{\vartheta(x)}{x}\leq \frac{(\log x)^2}{2\sqrt{x}\log 2}.</math>

This [[inequality (mathematics)|inequality]] implies that

:<math>\lim_{x\to\infty}\!\left(\frac{\psi(x)}{x}-\frac{\vartheta(x)}{x}\right)\! = 0.</math>

In other words, if one of the <math>\psi(x)/x</math> or <math>\vartheta(x)/x</math> tends to a [[limit of a function|limit]] then so does the other, and the two limits are equal.

'''Proof:''' Since <math>\psi(x)=\sum_{n \leq \log_2 x}\vartheta(x^{1/n})</math>, we find that

:<math>0 \leq \psi(x)-\vartheta(x)=\sum_{2\leq n \leq \log_2 x}\vartheta(x^{1/n}).</math>

But from the definition of <math>\vartheta(x)</math> we have the trivial inequality

:<math>\vartheta(x)\leq \sum_{p\leq x}\log x\leq x\log x</math>

so

:<math>\begin{align}
0\leq\psi(x)-\vartheta(x)&\leq \sum_{2\leq n\leq \log_2 x}x^{1/n}\log(x^{1/n})\\
&\leq(\log_2 x)\sqrt{x}\log\sqrt{x}\\
&=\frac{\log x}{\log 2}\frac{\sqrt{x}}{2}\log x\\
&=\frac{\sqrt{x}\,(\log x)^2}{2\log 2}.
\end{align}</math>

Lastly, divide by <math>x</math> to obtain the inequality in the theorem.

==Asymptotics and bounds==
The following bounds are known for the Chebyshev functions:{{ref|Dusart1999}}{{ref|Dusart2010}} (in these formulas {{math|''p''''k''}} is the {{mvar|k}}th prime number; {{math|''p''1 {{=}} 2}}, {{math|''p''2 {{=}} 3}}, etc.)

:<math>\begin{align}
\vartheta(p_k) &\ge k\left( \log k+\log\log k-1+\frac{\log\log k-2.050735}{\log k}\right)&& \text{for }k\ge10^{11}, \\[8px]
\vartheta(p_k) &\le k\left( \log k+\log\log k-1+\frac{\log\log k-2}{\log k}\right)&& \text{for }k \ge 198, \\[8px]
|\vartheta(x)-x| &\le 0.006788\,\frac{x}{\log x}&& \text{for }x \ge 10\,544\,111, \\[8px]
|\psi(x)-x|&\le0.006409\,\frac{x}{\log x}&& \text{for } x \ge e^{22},\\[8px]
0.9999\sqrt{x} &< \psi(x)-\vartheta(x)<1.00007\sqrt{x}+1.78\sqrt[3]{x}&& \text{for }x\ge121.
\end{align}</math>

Furthermore, under the [[Riemann hypothesis]],

:<math>\begin{align}
|\vartheta(x)-x| &= O\Big(x^{\frac12+\varepsilon}\Big) \\
|\psi(x)-x| &= O\Big(x^{\frac12+\varepsilon}\Big)
\end{align}</math>

for any {{math|''ε'' > 0}}.

Upper bounds exist for both {{math|''ϑ''&hairsp;&hairsp;(''x'')}} and {{math|''ψ''&hairsp;(''x'')}} such that<ref>{{Cite journal
| last1 = Rosser
| first1 = J. Barkley
| author-link1 = J. Barkley Rosser
| last2 = Schoenfeld
| first2 = Lowell
| author-link2 = Lowell Schoenfeld
| title = Approximate formulas for some functions of prime numbers.
| journal = Illinois J. Math.
| year = 1962
| volume = 6
| pages = 64–94
| url = http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ijm/1255631807}}</ref> {{ref|Dusart2010}}

:<math>\begin{align} \vartheta(x)&<1.000028x \\ \psi(x)&<1.03883x \end{align}</math>

for any {{math|''x'' > 0}}.

An explanation of the constant 1.03883 is given at {{OEIS2C|id=A206431}}.

==The exact formula==
In 1895, [[Hans Carl Friedrich von Mangoldt]] proved{{ref|Dav104}} an [[Explicit_formulae_(L-function)|explicit expression]] for {{math|''ψ''&hairsp;(''x'')}} as a sum over the nontrivial [[zero of a function|zeros]] of the [[Riemann zeta function]]:

:<math>\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \tfrac{1}{2} \log (1-x^{-2}).</math>

(The numerical value of {{math|{{sfrac|''ζ{{prime}} ''(0)|''ζ'' (0)}}}} is {{math|log(2π)}}.) Here {{mvar|ρ}} runs over the nontrivial zeros of the zeta function, and {{math|''ψ''0}} is the same as {{mvar|ψ}}, except that at its [[jump discontinuity|jump discontinuities]] (the prime powers) it takes the value halfway between the values to the left and the right:

:<math>\psi_0(x)
= \frac{1}{2}\!\left( \sum_{n \leq x} \Lambda(n)+\sum_{n < x} \Lambda(n)\right)
=\begin{cases} \psi(x) - \tfrac{1}{2} \Lambda(x) & x = 2,3,4,5,7,8,9,11,13,16,\dots \\ \,
\psi(x) & \mbox{otherwise.} \end{cases}</math>

From the [[Taylor series]] for the [[natural logarithm|logarithm]], the last term in the explicit formula can be understood as a summation of {{math|{{sfrac|''xω''|''ω''}}}} over the trivial zeros of the zeta function, {{math|''ω'' {{=}} −2, −4, −6, ...}}, i.e.

:<math>\sum_{k=1}^{\infty} \frac{x^{-2k}}{-2k} = \tfrac{1}{2} \log \left( 1 - x^{-2} \right).</math>

Similarly, the first term, {{math|''x'' {{=}} {{sfrac|''x''1|1}}}}, corresponds to the simple [[pole (complex analysis)|pole]] of the zeta function at 1. It being a pole rather than a zero accounts for the opposite sign of the term.

==Properties==
A theorem due to [[Erhard Schmidt]] states that, for some explicit positive constant {{mvar|K}}, there are infinitely many [[natural number]]s {{mvar|x}} such that

:<math>\psi(x)-x < -K\sqrt{x}</math>

and infinitely many natural numbers {{mvar|x}} such that

:<math>\psi(x)-x > K\sqrt{x}.</math>{{ref|Sch03}}{{ref|Hard16}}

In [[big-O notation|little-{{mvar|o}} notation]], one may write the above as

:<math>\psi(x)-x \ne o\left(\sqrt{x}\,\right).</math>

[[G. H. Hardy|Hardy]] and [[J. E. Littlewood|Littlewood]]{{ref|Hard16}} prove the stronger result, that

:<math>\psi(x)-x \ne o\left(\sqrt{x}\,\log\log\log x\right).</math>

==Relation to primorials==

The first Chebyshev function is the logarithm of the [[primorial]] of {{mvar|x}}, denoted {{math|''x''&hairsp;#}}:

:<math>\vartheta(x) = \sum_{p \le x} \log p = \log \prod_{p\le x} p = \log\left(x\#\right).</math>

This proves that the primorial {{math|''x''&hairsp;#}} is asymptotically equal to {{math|''e''(1&hairsp;&hairsp;+ ''o''(1))''x''}}, where "{{mvar|o}}" is the little-{{mvar|o}} notation (see [[Big O notation|big {{mvar|O}} notation]]) and together with the prime number theorem establishes the asymptotic behavior of {{math|''p''''n''&hairsp;#}}.

==Relation to the prime-counting function==
The Chebyshev function can be related to the prime-counting function as follows. Define

:<math>\Pi(x) = \sum_{n \leq x} \frac{\Lambda(n)}{\log n}.</math>

Then

:<math>\Pi(x) = \sum_{n \leq x} \Lambda(n) \int_n^x \frac{dt}{t \log^2 t} + \frac{1}{\log x} \sum_{n \leq x} \Lambda(n) = \int_2^x \frac{\psi(t)\, dt}{t \log^2 t} + \frac{\psi(x)}{\log x}.</math>

The transition from {{math|Π}} to the [[prime-counting function]], {{mvar|π}}, is made through the equation

:<math>\Pi(x) = \pi(x) + \tfrac{1}{2} \pi\left(\sqrt{x}\,\right) + \tfrac{1}{3} \pi\left(\sqrt[3]{x}\,\right) + \cdots</math>

Certainly {{math|''π''&hairsp;(''x'') ≤ ''x''}}, so for the sake of approximation, this last relation can be recast in the form

:<math>\pi(x) = \Pi(x) + O\left(\sqrt{x}\,\right).</math>

==The Riemann hypothesis==
The [[Riemann hypothesis]] states that all nontrivial [[zero of a function|zeros]] of the zeta function have [[real part]] {{sfrac|1|2}}. In this case, {{math|{{abs|''x''&hairsp;''ρ''}} {{=}} {{sqrt|''x''}}}}, and it can be shown that

:<math>\sum_{\rho} \frac{x^{\rho}}{\rho} = O\!\left(\sqrt{x}\, \log^2 x\right).</math>

By the above, this implies

:<math>\pi(x) = \operatorname{li}(x) + O\!\left(\sqrt{x}\, \log x\right).</math>

==Smoothing function==
[[Image:Chebyshev-smooth.svg|thumb|right|The difference of the smoothed Chebyshev function and {{math|{{sfrac|''x''&hairsp;2|2}}}}
for {{math|''x'' < 106}}]]

The '''smoothing function''' is defined as

:<math>\psi_1(x) = \int_0^x \psi(t)\,dt.</math>

Obviously <math>\psi_1(x) \sim \frac{x^2}{2}.</math>

==Notes==
<references/>
* {{note|Dusart2010}} [[Pierre Dusart]], "Estimates of some functions over primes without R.H.". {{arxiv|1002.0442}}
* {{note|Dusart1999}} Pierre Dusart, "Sharper bounds for {{mvar|ψ}}, {{mvar|θ}}, {{mvar|π}}, {{math|''p''''k''}}", Rapport de recherche no. 1998-06, Université de Limoges. An abbreviated version appeared as "The {{math|''k''}}th prime is greater than {{math|''k''(log ''k'' + log log ''k'' − 1)}} for {{math|''k'' ≥ 2}}", ''Mathematics of Computation'', Vol. 68, No. 225 (1999), pp. 411–415.
* {{note|Sch03}}Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", ''Mathematische Annalen'', '''57''' (1903), pp. 195–204.
* {{note|Hard16}}G .H. Hardy and J. E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", ''Acta Mathematica'', '''41''' (1916) pp. 119–196.
* {{note|Dav104}}[[Harold Davenport|Davenport, Harold]] (2000). In ''[https://books.google.com/books?id=U91lsCaJJmsC&pg=PA104 Multiplicative Number Theory]''. Springer. p. 104. {{isbn|0-387-95097-4}}. Google Book Search.

==References==
* {{Apostol IANT}}

==External links==
* {{mathworld|urlname=ChebyshevFunctions|title=Chebyshev functions}}
* {{planetmathref| urlname=MangoldtSummatoryFunction| title=Mangoldt summatory function}}
* {{planetmathref| urlname=ChebyshevFunctions| title=Chebyshev functions}}
* [http://www.math.ucsb.edu/~stopple/explicit.html Riemann's Explicit Formula], with images and movies

[[Category:Arithmetic functions]]

Chebyshev function - Revision history

imported>Lumidek: /* The exact formula */ spacing \, instead of [5px] which shows as "[5px]" in CHTML MathJax rendering