Chebyshev function

The Chebyshev function Template:Math, with Template:Math

File:Chebyshev.svg

The function Template:Math, for Template:Math

File:Chebyshev-big.svg

The function Template:Math, for Template:Math

In mathematics, the Chebyshev function is either of two related functions. The first Chebyshev function Template:Math or Template:Math is given by

$\vartheta (x)=\sum _{p\leq x}\log p$

with the sum extending over all prime numbers Template:Mvar that are less than or equal to Template:Mvar.

The second Chebyshev function Template:Math is defined similarly, with the sum extending over all prime powers not exceeding Template:Mvar:

$\psi (x)=\sum _{p^{k}\leq x}\log p=\sum _{n\leq x}\Lambda (n)=\sum _{p\leq x}\left\lfloor \log _{p}x\right\rfloor \log p,$

where Template:Mvar is the von Mangoldt function. The Chebyshev functions, especially the second one Template:Math, are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, Template:Math (See the exact formula, below.) Both Chebyshev functions are asymptotic to Template:Mvar, 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

$\psi (x)=\sum _{p\leq x}k\log p$

where Template:Mvar is the unique integer such that Template:Math and Template:Math. The values of Template:Mvar are given in Template:OEIS2C. A more direct relationship is given by

$\psi (x)=\sum _{n=1}^{\infty }\vartheta \left(x^{\frac {1}{n}}\right).$

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

$\vartheta \left(x^{\frac {1}{n}}\right)=0\quad {\text{for}}\quad n>\log _{2}x\ ={\frac {\log x}{\log 2}}.$

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

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

Values of Template:Math for the integer variable Template:Mvar is given at Template:OEIS2C.

Asymptotics and bounds

The following bounds are known for the Chebyshev functions:Template:RefTemplate:Ref (in these formulas Template:Math is the Template:Mvarth prime number Template:Math, Template:Math, etc.)

${\begin{aligned}\vartheta (p_{k})&\geq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2.050735}{\ln k}}\right)&&{\text{for }}k\geq 10^{11},\\[8px]\vartheta (p_{k})&\leq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2}{\ln k}}\right)&&{\text{for }}k\geq 198,\\[8px]|\vartheta (x)-x|&\leq 0.006788{\frac {x}{\ln x}}&&{\text{for }}x\geq 10\,544\,111,\\[8px]|\psi (x)-x|&\leq 0.006409{\frac {x}{\ln x}}&&{\text{for }}x\geq e^{22},\\[8px]0.9999{\sqrt {x}}&<\psi (x)-\vartheta (x)<1.00007{\sqrt {x}}+1.78{\sqrt[{3}]{x}}&&{\text{for }}x\geq 121.\end{aligned}}$

Furthermore, under the Riemann hypothesis,

${\begin{aligned}|\vartheta (x)-x|&=O\left(x^{{\frac {1}{2}}+\varepsilon }\right)\\|\psi (x)-x|&=O\left(x^{{\frac {1}{2}}+\varepsilon }\right)\end{aligned}}$

for any Template:Math.

Upper bounds exist for both Template:Math and Template:Math such that, Template:Ref

${\begin{aligned}\vartheta (x)&<1.000028x\\\psi (x)&<1.03883x\end{aligned}}$

for any Template:Math.

An explanation of the constant 1.03883 is given at Template:OEIS2C.

The exact formula

In 1895, Hans Carl Friedrich von Mangoldt provedTemplate:Ref an explicit expression for Template:Math as a sum over the nontrivial zeros of the Riemann zeta function:

$\psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-{\frac {\zeta '(0)}{\zeta (0)}}-{\tfrac {1}{2}}\log(1-x^{-2}).$

(The numerical value of Template:Math is Template:Math.) Here Template:Mvar runs over the nontrivial zeros of the zeta function, and Template:Math is the same as Template:Mvar, except that at its jump discontinuities (the prime powers) it takes the value halfway between the values to the left and the right:

$\psi _{0}(x)={\tfrac {1}{2}}\left(\sum _{n\leq x}\Lambda (n)+\sum _{n<x}\Lambda (n)\right)={\begin{cases}\psi (x)-{\tfrac {1}{2}}\Lambda (x)&x=2,3,4,5,7,8,9,11,13,16,\dots \\[5px]\psi (x)&{\mbox{otherwise.}}\end{cases}}$

From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of Template:Math over the trivial zeros of the zeta function, Template:Math, i.e.

$\sum _{k=1}^{\infty }{\frac {x^{-2k}}{-2k}}={\tfrac {1}{2}}\log \left(1-x^{-2}\right).$

Similarly, the first term, Template:Math, corresponds to the simple 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 Template:Mvar, there are infinitely many natural numbers Template:Mvar such that

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

and infinitely many natural numbers Template:Mvar such that

$\psi (x)-x>K{\sqrt {x}}.$Template:RefTemplate:Ref

In [[big-O notation|little-Template:Mvar notation]], one may write the above as

$\psi (x)-x\neq o\left({\sqrt {x}}\right).$

Hardy and LittlewoodTemplate:Ref prove the stronger result, that

$\psi (x)-x\neq o\left({\sqrt {x}}\log \log \log x\right).$

Relation to primorials

The first Chebyshev function is the logarithm of the primorial of Template:Mvar, denoted Template:Math:

$\vartheta (x)=\sum _{p\leq x}\log p=\log \prod _{p\leq x}p=\log \left(x\#\right).$

This proves that the primorial Template:Math is asymptotically equal to Template:Math, where "Template:Mvar" is the little-Template:Mvar notation (see [[Big O notation|big Template:Mvar notation]]) and together with the prime number theorem establishes the asymptotic behavior of Template:Math.

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 Template:Mvar to the prime-counting function, Template:Mvar, is made through the equation

$\Pi (x)=\pi (x)+{\tfrac {1}{2}}\pi \left({\sqrt {x}}\right)+{\tfrac {1}{3}}\pi \left({\sqrt[{3}]{x}}\right)+\cdots$

Certainly Template:Math, so for the sake of approximation, this last relation can be recast in the form

$\pi (x)=\Pi (x)+O\left({\sqrt {x}}\right).$

The Riemann hypothesis

The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part Template:Sfrac. In this case, Template:Math, and it can be shown that

$\sum _{\rho }{\frac {x^{\rho }}{\rho }}=O\left({\sqrt {x}}\log ^{2}x\right).$

By the above, this implies

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

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 Template:Mvar we get Template:Math. Manipulating, we have the "Trace formula" for the exponential of the Hamiltonian operator satisfying

$\left.\zeta \left({\tfrac {1}{2}}+i{\hat {H}}\right)\right|n\geq \zeta \left({\tfrac {1}{2}}+iE_{n}\right)=0,$

and

$\sum _{n}e^{iuE_{n}}=Z(u)=e^{\frac {u}{2}}-e^{-{\frac {u}{2}}}{\frac {d\psi _{0}}{du}}-{\frac {e^{\frac {u}{2}}}{e^{3u}-e^{u}}}=\operatorname {Tr} \left(e^{iu{\hat {H}}}\right),$

where the "trigonometric sum" can be considered to be the trace of the operator (statistical mechanics) Template:Math, which is only true if Template:Math.

Using the semiclassical approach the potential of Template:Math satisfies:

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

with Template:Math as Template:Math.

solution to this nonlinear integral equation can be obtained (among others) by

$V^{-1}(x)\approx {\sqrt {4\pi }}\cdot {\frac {d^{\frac {1}{2}}}{dx^{\frac {1}{2}}}}N(x)$

in order to obtain the inverse of the potential:

$\pi N(E)=\operatorname {Arg} \xi \left({\tfrac {1}{2}}+iE\right).$

Smoothing function

File:Chebyshev-smooth.svg

The difference of the smoothed Chebyshev function and Template:Math for Template:Math

The smoothing function is defined as

$\psi _{1}(x)=\int _{0}^{x}\psi (t)\,dt.$

It can be shown that

$\psi _{1}(x)\sim {\frac {x^{2}}{2}}.$

Variational formulation

The Chebyshev function evaluated at Template:Math 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

$f(t)=\psi \left(e^{t}\right)e^{-ct}\quad {\text{for }}c>0.$

Notes

• Template:Note Pierre Dusart, "Estimates of some functions over primes without R.H.". Template:Arxiv
• Template:Note Pierre Dusart, "Sharper bounds for Template:Mvar, Template:Mvar, Template:Mvar, Template:Math", Rapport de recherche no. 1998-06, Université de Limoges. An abbreviated version appeared as "The Template:Mathth prime is greater than Template:Math for Template:Math", Mathematics of Computation, Vol. 68, No. 225 (1999), pp. 411–415.
• Template:NoteErhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", Mathematische Annalen, 57 (1903), pp. 195–204.
• Template:NoteG .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.
• Template:NoteDavenport, Harold (2000). In Multiplicative Number Theory. Springer. p. 104. Template:Isbn. Google Book Search.

References

• Template:Apostol IANT

External links

• Template:Mathworld
• Template:Planetmathref
• Template:Planetmathref
• Riemann's Explicit Formula, with images and movies