Chebyshev function

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The Chebyshev function

ψ(x)

, with

x < 50

The Chebyshev function

ψ(x) - x

, for

x < 10,000

The Chebyshev function

ψ(x) - x

, for $x < 10\ \mathrm{million}$

The Chebyshev function is either of two related functions. The first Chebyshev function ϑ(x) or θ(x) is given by

$\vartheta(x)=\sum_{p\le x} \log p$

with the sum extending over all prime numbers p that are less than x. The second Chebyshev function $ψ(x)$ is defined by

$\psi(x) = \sum_{n \leq x} \Lambda(n),$

where $Λ$ is the von Mangoldt function. The Chebyshev function is often used in proofs related to prime numbers, because it is typically simpler to work with than the prime counting function, $π(x)$ . Both functions are asymptotic to $x$ , a statement equivalent to the prime number theorem.

Both functions are named in honour of Pafnuty Lvovich Chebyshev.

1 Relationships
2 Asymptotics and bounds
3 The exact formula
4 Properties
5 Relation to primorials
6 Relation to the prime counting function
7 The Riemann hypothesis
8 Smoothing function
9 Variational formulation
10 References

[edit] Relationships

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

$\psi(x)=\sum_{p\le x} k \log p$

where k is the unique integer such that $p^k\le x$ but $p k + 1 > x$ . A more direct relationship is given by

$\psi(x)=\sum_{n=1}^\infty \vartheta \left(x^{1/n}\right).$

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

$\vartheta \left(x^{1/n}\right) = 0$ for

n > log 2 x .

[edit] Asymptotics and bounds

Pierre Dusart^[1] proved the following bounds for the Chebyshev functions:

$\vartheta(p_k)\ge k\left( \ln k+\ln\ln k-1+\frac{\ln\ln k-2.0553}{\ln k}\right)$ for k ≥ exp(22)

$\vartheta(p_k)\le k\left( \ln k+\ln\ln k-1+\frac{\ln\ln k-2}{\ln k}\right)$ for k ≥ 198

$\psi(p_k)\le k\left( \ln k+\ln\ln k-1+\frac{\ln\ln k-2}{\ln k}\right) + 1.43\sqrt x$ for k ≥ 198

$|\vartheta(x)-x|\le0.006788\frac{x}{\ln x}$ for x ≥ 10,544,111

$|\psi(x)-x|\le0.006409\frac{x}{\ln x}$ for x ≥ exp(22)

$\psi(x)-\vartheta(x)<0.0000132\frac{x}{\ln x}$ for x ≥ exp(30)

Along with $\psi(x)\ge \vartheta(x)$ , this gives a good characterization of the behavior of these two functions.

[edit] The exact formula

In 1895, Hans Carl Friedrich von Mangoldt proved^[2] an explicit expression for $ψ(x)$ as a sum over the nontrivial zeros of the Riemann zeta function:

$\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2}).$

Here $ρ$ runs over the nontrivial zeros of the zeta function, and

$\psi_0(x) = \begin{cases} \psi(x) - \frac{1}{2} \Lambda(x) & x = p^m \mbox{, p prime, m an integer} \\ \psi(x) & \mbox{otherwise.} \end{cases}$

From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of $- x ω / ω$ over the trivial zeros of the zeta function, $\omega = -2, -4, -6, \cdots$ , i.e.

$\sum_{k=1}^{\infty} \frac{x^{2k}}{2k} = \frac{1}{2} \log ( 1 - x^{-2} ).$

[edit] Properties

A theorem due to Erhard Schmidt states that, for any real, positive K, there are values of x such that

$\psi(x)-x < -K\sqrt{x}$

and

$\psi(x)-x > K\sqrt{x}$

infinitely often^[3]^[4]. On big-O notation, one may write the above as

$\psi(x)-x \ne O\left(\sqrt{x}\right).$

Hardy and Littlewood^[5] prove the stronger result, that

$\psi(x)-x \ne O\left(\sqrt{x}\log\log\log x\right).$

[edit] Relation to primorials

The first Chebyshev function is the logarithm of the primorial of x, denoted x#:

$\vartheta(x)=\sum_{p\le x} \log p=\log \prod_{p\le x} p = \log x\#.$

This proves that the primorial x# is asymptotically equal to exp((1+o(1))x), where "o" is the little-o notation (see Big O notation) and together with the prime number theorem establishes the asymptotic behavior of p_n#.

[edit] Relation to the prime counting function

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

$\pi_1(x) = \sum_{n \leq x} \frac{\Lambda(n)}{\log n}$

Then

$\pi_1(x) = \sum_{n \leq x} \Lambda(n) \int_n^x \frac{dt}{t \log^2 t} + \frac{1}{\log x} \sum_{n \leq x} \Lambda(n) = \int_2^x \frac{\psi(t) dt}{t \log^2 t} + \frac{\psi(x)}{\log x}$ .

The transition from $π 1$ to the prime counting function, $π$ , is made through the equation

$\pi_1(x) = \pi(x) + \frac{1}{2} \pi(x^{1/2}) + \frac{1}{3} \pi(x^{1/3}) + \cdots$

Certainly $\pi(x) \leq x$ , so for the sake of approximation, this last relation can be recast in the form

$\pi(x) = \pi_1(x) + O(\sqrt x)$ .

[edit] The Riemann hypothesis

The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part 1/2. In this case, $|x^{\rho}|=\sqrt x$ , and it can be shown that

$\sum_{\rho} \frac{x^{\rho}}{\rho} = O(\sqrt x \log^2 x)$ .

By the above, this implies

$\pi(x) = \operatorname{li}(x) + O(\sqrt x \log x).$

A good evidence that RH could be true comes from the fact proposed by Alain Connes and others, that if we differentiate V. Mangoldt formula respect to x make x=exp(u) manipulating we have the formula we have the "Trace formula" for the exponential of the Hamiltonian operator satisfying

$\zeta(1/2+i \hat H )|n>=\zeta(1/2+iE_{n})=0$

$\sum_{n}e^{iu E_{n}}=Z(u)=e^{u/2}-e^{-u/2} \frac{d\psi _{0}}{du}-\frac{e^{u/2}}{e^{3u}-e^{u}}=Tr(e^{iu\hat H })$

Where the "Trigonometric sum" can be considered to be the trace of the operator (Statistical Mechanics) $e^{iu \hat H}$ ,which is only true if $ρ = 1 / 2 + i E (n)$ .

Using the semiclassical approach the potential of H=T+V satisfies:

$\frac{Z(u)u^{1/2}}{\sqrt \pi }\sim \int_{-\infty}^{\infty}dxe^{i (uV(x)+ \pi /4 )}$ with Z(u) is 0 as u tends to oo.

[edit] Smoothing function

The smoothed Chebyshev function

ψ 1 (x) - x 2 / 2

, for

x < 10 6

The smoothing function is defined as

$\psi_1(x)=\int_0^x \psi(t)\,dt.$

It can be shown that

$\psi_1(x) \sim \frac{x^2}{2}.$

[edit] Variational formulation

The Chebyshev function evaluated at x = exp(t) minimizes the functional

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

$f(t)= \psi (e^t)e^{-ct},\,$

for c > 0.

[edit] References

↑ Pierre Dusart, "Sharper bounds for ψ, θ, π, $p k$ ", Rapport de recherche n° 1998-06, Université de Limoges. An abbreviated version appeared as "The kth prime is greater than k(ln k + ln ln k - 1) for k ≥ 2", Mathematics of Computation, Vol. 68, No. 225 (1999), pp. 411&dash;415.
↑ Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", Mathematische Annalen, 57 (1903), pp.195-204.
↑ 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.
↑ Davenport, Harold (2000). In Multiplicative Number Theory. Springer. p. 104. ISBN 0-387-95097-4. Google Book Search.
J. Garcia “Chebyshev Statistical Partition function: A connection between Statistical Mechanics and Riemann Hypothesis” Annals of Mathematics (2005) J. Hopkins Univ. Presss ISSN 0003-486X 'e-print ' [6]
Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9
Eric W. Weisstein, Chebyshev functions at MathWorld.
Mangoldt summatory function on PlanetMath
Chebyshev functions on PlanetMath
Riemann's Explicit Formula, with images and movies