Chebyshev function

From Encyclopedia of Mathematics
Revision as of 17:01, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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 .


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


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


[a1] A. Ivic, "The Riemann zeta-function" , Wiley (1985)
How to Cite This Entry:
Chebyshev function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article