# Chebyshev function

From Encyclopedia of Mathematics

One of the two functions, of a positive argument , defined as follows:

The first sum is taken over all prime numbers , and the second over all positive integer powers of prime numbers such that . The function can be expressed in terms of the Mangoldt function

It follows from the definitions of and that is equal to the product of all prime numbers , and that the quantity is equal to the least common multiple of all positive integers . The functions and are related by the identity

These functions are also closely connected with the function

which expresses the number of the prime numbers .

#### References

[1] | P.L. Chebyshev, "Mémoire sur les nombres premiers" J. Math. Pures Appl. , 17 (1852) pp. 366–390 (Oeuvres, Vol. 1, pp. 51–70) |

#### Comments

For properties of the Chebyshev functions and see [a1], Chapt. 12.

#### References

[a1] | A. Ivic, "The Riemann zeta-function" , Wiley (1985) |

**How to Cite This Entry:**

Chebyshev function.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_function&oldid=12881

This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article