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The Chebyshev function ψ(x), with x < 50

The function ψ(x) − x, for x < 10,000

The function ψ(x) − x, for x < 10 million

In mathematics, 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 or equal to x.

The second Chebyshev function ψ(x) is defined similarly, with the sum extending over all prime powers not exceeding x:

$\psi(x) = \sum_{p^k\le x}\log p=\sum_{n \leq x} \Lambda(n) = \sum_{p\le x}\lfloor\log_p x\rfloor\log p,$

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 Chebyshev.

[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 ≤ 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

$\vartheta \left(x^{1/n}\right) = 0\text{ for }n>\log_2 x\,.$

The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to n.

$\operatorname{lcm}(1,2,\dots, n)=e^{\psi(n)}.$

[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 ψ₀ is the same as ψ, except that at its jump discontinuities (the prime powers) it takes the value halfway between the values to the left and the right:

$\psi_0(x) = \frac12\left( \sum_{n \leq x} \Lambda(n)+\sum_{n < x} \Lambda(n)\right) =\begin{cases} \psi(x) - \frac{1}{2} \Lambda(x) & x = 2,3,4,5,7,8,9,11,13,16,\dots \\ \psi(x) & \mbox{otherwise.} \end{cases}$

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, $\omega = -2, -4, -6, \ldots$ , 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

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

Then

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

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

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

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

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

[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

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

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

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

$\sum_n e^{iu E_{n}}=Z(u) = e^{u/2}-e^{-u/2} \frac{d\psi _0}{du}-\frac{e^{u/2}}{e^{3u}-e^u} = \operatorname{Tr}(e^{iu\hat H }),$

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:

$\frac{Z(u)u^{1/2}}{\sqrt \pi }\sim \int_{-\infty}^\infty e^{i (uV(x)+ \pi /4 )}\,dx$

with Z(u) → 0 as u → ∞.

[edit] Smoothing function

The smoothed Chebyshev function ψ₁(x) − x²/2, for x < 10⁶

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

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

so

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

for c > 0.

[edit] Notes

^ 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. 411–415.
^ 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.

[edit] References

Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, MR 0434929, ISBN 978-0-387-90163-3

[edit] External links

Weisstein, Eric W., "Chebyshev functions" from MathWorld.
Mangoldt summatory function on PlanetMath
Chebyshev functions on PlanetMath
Riemann's Explicit Formula, with images and movies

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