Chebyshev function

The Chebyshev function, for example in English and Chebyshev function or similarly named, is one of two number-theoretic functions that are named after the Russian mathematician Chebyshev Pafnuti Lvovitch. You get through their connection with the Primzahlzählfunktion and the prime number theorem and thus the Riemann zeta function in importance.

The first Chebyshev function, commonly referred to, or the sum of the logarithms of primes to:

The second Chebyshev function is a function of the summed Mangoldt function:

Wherein the function is defined as Mangoldt

Basic Properties

The first Chebyshev function can also be represented as

The Primfakultät called.

The second can also be written as the logarithm of the least common multiples of 1 to:

According to Erhard Schmidt is available for every positive real values ​​, so

And

Infinitely often.

Asymptotics

It is

That is,

Also applies

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

Relationship between the two functions

It is

Being uniquely determined by and is completely and then.

A more direct connection is created by

Note that for

The " exact formula "

1895 proved Hans Karl Friedrich von Mangoldt following formula, which is also referred to as "explicit formula" in English:

It is not prime or a prime power and the sum runs over all non-trivial zeros of the Riemann zeta function.

Credentials

• 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 9,780,387,950,976th §. 17 GBS, restricted