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[[Image:ChebyshevPsi.png|thumb|right|The Chebyshev function {{math|''ψ''(''x'')}}, with {{math|''x'' < 50}}]]
[[Image:Chebyshev.svg|thumb|right|The function {{math|''ψ''(''x'') − ''x''}}, for {{math|''x'' < 10<sup>4</sup>}}]]
[[Image:Chebyshev-big.svg|thumb|right|The function {{math|''ψ''(''x'') − ''x''}}, for {{math|''x'' < 10<sup>7</sup>}}]]

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

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

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''(''x'')}} is defined similarly, with the sum extending over all prime powers not exceeding&nbsp;{{mvar|x}}:

:<math> \psi(x) = \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 {{mvar|Λ}} is the [[von Mangoldt function]]. The Chebyshev functions, especially the second one {{math|''ψ''(''x'')}}, are often used in proofs related to [[prime numbers]], because it is typically simpler to work with them than with the [[prime-counting function]], {{math|''π''(''x'')}} (See [[#The exact formula|the exact formula]], below.) Both Chebyshev functions are asymptotic to&nbsp;{{mvar|x}}, a statement equivalent to the [[prime number theorem]].

Both 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''<sup>''k''</sup> ≤ ''x''}} and {{math|''x'' < ''p''<sup>''k'' + 1</sup>}}. 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 \left(x^\frac{1}{n}\right).</math>

Note that this last sum has only a finite number of non-vanishing terms, as

:<math>\vartheta \left(x^\frac{1}{n}\right) = 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&nbsp;{{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}} is given at {{OEIS2C|id=A003418}}.

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

:<math>\begin{align}
\vartheta(p_k)&\ge k\left( \ln k+\ln\ln k-1+\frac{\ln\ln k-2.050735}{\ln k}\right)&& \text{for }k\ge10^{11}, \\[8px]
\vartheta(p_k)&\le k\left( \ln k+\ln\ln k-1+\frac{\ln\ln k-2}{\ln k}\right)&& \text{for }k \ge 198, \\[8px]
|\vartheta(x)-x|&\le0.006788\frac{x}{\ln x}&& \text{for }x \ge 10\,544\,111, \\[8px]
|\psi(x)-x|&\le0.006409\frac{x}{\ln 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\left(x^{\frac12+\varepsilon}\right) \\ |\psi(x)-x|&=O\left(x^{\frac12+\varepsilon}\right) \end{align}</math>

for any {{math|''ε'' > 0}}.

Upper bounds exist for both {{math|''ϑ''(''x'')}} and {{math|''ψ''(''x'')}} such that,<ref>{{Cite journal
  | last =  Rosser
  | first = 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''(''x'')}} as a sum over the nontrivial 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′''(0)|''ζ''(0)}}}} is {{math|log(2π)}}.) Here {{mvar|ρ}} runs over the nontrivial zeros of the zeta function, and {{math|''ψ''<sub>0</sub>}} is the same as {{mvar|ψ}}, except that at its jump discontinuities (the prime powers) it takes the value halfway between the values to the left and the right:

: <math> \psi_0(x) 
= \tfrac12\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 \\ [5px]
\psi(x) & \mbox{otherwise.} \end{cases}</math>

From the [[Taylor series]] for the [[logarithm]], the last term in the explicit formula can be understood as a summation of {{math|{{sfrac|''x<sup>ω</sup>''|''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''<sup>1</sup>|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 numbers {{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''#}}:

:<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''#}} is asymptotically equal to {{math|''e''<sup>(1 + ''o''(1))''x''</sup>}}, 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''<sub>''n''</sub>#}}.

==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 {{mvar|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''(''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 zeros of the zeta function have real part {{sfrac|1|2}}. In this case, {{math|{{abs|''x<sup>ρ</sup>''}} {{=}} {{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>

Good evidence that the hypothesis could be true comes from the fact proposed by [[Alain Connes]] and others, that if we differentiate the von Mangoldt formula with respect to {{mvar|x}} we get {{math|''x'' {{=}} ''e<sup>u</sup>''}}. Manipulating, we have the "Trace formula" for the exponential of the Hamiltonian operator satisfying

: <math>\left. \zeta\left(\tfrac12+i \hat H \right)\right|n \ge \zeta\left(\tfrac12+iE_{n}\right)=0, </math>
and
:<math> \sum_n e^{iu E_{n}}=Z(u) = e^\frac{u}{2}-e^{-\frac{u}{2}} \frac{d\psi _0}{du}-\frac{e^\frac{u}{2}}{e^{3u}-e^u} = \operatorname{Tr}\left(e^{iu\hat H }\right),</math>

where the "trigonometric sum" can be considered to be the trace of the operator ([[statistical mechanics]]) {{math|''e<sup>iuĤ</sup>''}}, which is only true if {{math|''ρ'' {{=}} {{sfrac|1|2}} + ''iE''(''n'')}}.

Using the semiclassical approach the potential of {{math|''H'' {{=}} ''T'' + ''V''}} satisfies:

:<math> \frac{Z(u)u^\frac12}{\sqrt \pi }\sim \int_{-\infty}^\infty e^{i \left(uV(x)+ \frac{\pi}{4} \right)}\,dx </math>

with {{math|''Z''(''u'') → 0}} as&nbsp;{{math|''u'' → ∞}}.

solution to this nonlinear integral equation can be obtained (among others) by
:<math>V^{-1} (x) \approx \sqrt {4\pi}\cdot \frac{d^\frac12}{dx^\frac12} N(x) </math>
in order to obtain the inverse of the potential:
:<math> \pi N(E) = \operatorname{Arg} \xi \left(\tfrac12+iE\right). </math>

==Smoothing function==
[[Image:Chebyshev-smooth.svg|thumb|right|The difference of the smoothed Chebyshev function and {{math|{{sfrac|''x''<sup>2</sup>|2}}}} 
for {{math|''x'' < 10<sup>6</sup>}}]]

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

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

It can be shown that

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

==Variational formulation==

The Chebyshev function evaluated at {{math|''x'' {{=}} ''e<sup>t</sup>''}} minimizes the functional

:<math> J[f]=\int_{0}^{\infty}\frac{f(s)\zeta' (s+c)}{\zeta(s+c)(s+c)}\,ds-\int_{0}^{\infty}\!\!\!\int_{0}^{\infty} e^{-st}f(s)f(t)\,ds\,dt, </math>

so

:<math> f(t)= \psi \left(e^t\right)e^{-ct} \quad\text{for } c > 0.</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''<sub>''k''</sub>}}", Rapport de recherche no. 1998-06, Université de Limoges.  An abbreviated version appeared as "The {{math|''k''}}th prime is greater than {{math|''k''(ln ''k'' + ln ln ''k'' − 1)}} for {{math|''k'' ≥ 2}}", ''Mathematics of Computation'', Vol. 68, No. 225 (1999), pp.&nbsp;411&ndash;415.
* {{note|Sch03}}Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", ''Mathematische Annalen'', '''57''' (1903), pp.&nbsp;195&ndash;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.&nbsp;119&ndash;196.
* {{note|Dav104}}[[Harold Davenport|Davenport, Harold]] (2000). In ''[https://books.google.com/books?vid=ISBN0387950974&id=U91lsCaJJmsC&pg=PA104&lpg=PA104 Multiplicative Number Theory]''. Springer.  p.&nbsp;104. {{isbn|0-387-95097-4}}. Google Book Search.

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

==External links==
* {{mathworld|urlname=ChebyshevFunctions|title=Chebyshev functions}}
* {{planetmathref|id=4020|title=Mangoldt summatory function}}
* {{planetmathref|id=4573|title=Chebyshev functions}}
* [http://www.math.ucsb.edu/~stopple/explicit.html Riemann's Explicit Formula], with images and movies

[[Category:Arithmetic functions]]