Chebyshev function

The Chebyshev function , also known as Chebyshev function or similar, is one of two number-theoretic functions named after the Russian mathematician Pafnuti Lwowitsch Chebyshev . They gain in importance due to their connection with the prime number counting function and the prime number theorem, and thus the Riemann zeta function .

The first Chebyshev function , usually labeled or , is the sum of the logarithms of the prime numbers up to : ${\displaystyle \theta \,}$ ${\displaystyle \vartheta }$ ${\displaystyle x}$

${\displaystyle \vartheta (x)=\sum _{p\leq x \atop p{\text{ prim}}}\operatorname {log} (p)}$

The second Chebyshev function is the summed function of the Mangoldt function : ${\displaystyle \psi (x)}$

${\displaystyle \psi (x)=\sum _{n=1}^{x}\Lambda (n)=\sum _{p^{k}\leq x}\operatorname {log} (p)}$

where the Mangoldt function is defined as ${\displaystyle \Lambda }$

${\displaystyle \Lambda (n)={\begin{cases}\log(p)&{\text{falls }}n{\text{ sich als }}n=p^{k}{\text{ darstellen lässt, wobei }}p{\text{ prim, }}k\in \mathbb {N} ^{+}\\0&{\text{sonst}}\end{cases}}}$

Basic properties

The former Chebyshev function can also be represented as

${\displaystyle \vartheta (x)=\log(x_{\#}),}$

where denotes the prime faculty . ${\displaystyle x_{\#}}$

The second can also be written as the logarithm of the least common multiple from 1 to : ${\displaystyle n}$

${\displaystyle \psi (x)=\operatorname {log} (\operatorname {kgV} (1,2,3,\ldots ,\lfloor x\rfloor ))}$

According to Erhard Schmidt, there are positive real values ​​for every so that ${\displaystyle k}$ ${\displaystyle x}$

${\displaystyle \psi (x)-x<-k{\sqrt {x}}}$

and

${\displaystyle \psi (x)-x>k{\sqrt {x}}}$

infinitely often.

Asymptotics

It applies

${\displaystyle \lim _{x\to \infty }{\frac {x}{\vartheta (x)}}=1,}$

d. H.

${\displaystyle \vartheta (n)\sim n.}$

The same applies

${\displaystyle \psi (n)\sim n.\,}$

Pierre Dusart found a number of barriers to the two functions:

${\displaystyle \vartheta (p_{k})\geq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2{,}0553}{\ln k}}\right),\qquad k\geq \exp(22)}$

${\displaystyle \vartheta (p_{k})\leq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2}{\ln k}}\right),\qquad k\geq 198}$

${\displaystyle \psi (p_{k})\leq k\left(\ln k+\ln \ln k-1+{\frac {\ln \ln k-2}{\ln k}}\right)+1{,}43{\sqrt {x}},\qquad k\geq 198}$

${\displaystyle |\vartheta (x)-x|\leq 0{,}006788\,{\frac {x}{\ln x}},\qquad x\geq 10{.}544{.}111}$

${\displaystyle |\psi (x)-x|\leq 0{,}006409\,{\frac {x}{\ln x}},\qquad x\geq \exp(22)}$

${\displaystyle \psi (x)-\vartheta (x)<0{,}0000132\,{\frac {x}{\ln x}},\qquad x\geq \exp(30).}$

Relationship between the two functions

It applies

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

where whole and then through and uniquely is determined. ${\displaystyle k}$ ${\displaystyle p^{k}\leq x}$ ${\displaystyle p^{k+1}\geq x}$

A more direct connection is created by

${\displaystyle \psi (x)=\sum _{n=1}^{\infty }\vartheta \left(x^{\frac {1}{n}}\right)=\sum _{n=1}^{\lfloor \log _{2}(x)\rfloor }\vartheta \left(x^{\frac {1}{n}}\right).}$

Note that for ${\displaystyle \vartheta \left(x^{\frac {1}{n}}\right)=0}$ ${\displaystyle n\geq \log _{2}(x).}$

The "exact formula"

In 1895 Hans Karl Friedrich von Mangoldt proved the following formula, which is also called "explicit formula" in English :

${\displaystyle \psi (x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\ln(2\pi )-{\frac {1}{2}}\ln \left(1-x^{-2}\right)}$

Here and is not prime or a prime power and the sum runs over all nontrivial zeros of the Riemann zeta function . ${\displaystyle x>1}$ ${\displaystyle \rho \,}$ ${\displaystyle \zeta }$

credentials

1. ^ Pierre Dusart: Sharper bounds for ψ, θ, π, p _k . In: Rapport de recherche n ° 1998-06, Université de Limoges. PDF
2. ↑ Eric W. Weisstein : Explicit Formula . In: MathWorld (English).

• Eric W. Weisstein : Chebyshev Function . In: MathWorld (English).
• Mangoldt Summatory Function and Chebyshev Function on PlanetMath
• Harold Davenport , Hugh L. Montgomery : Multiplicative number theory . Springer Verlag 2000, ISBN 0387950974 , ISBN 9780387950976 . §. 17. GBS, restricted

Web links

Wikiversity: Chebyshev's Assessments  - Course Materials