Chebyshev function
From Encyclopedia of Mathematics
One of the two functions, of a positive argument , defined as follows:
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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
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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
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These functions are also closely connected with the function
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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=33827
Chebyshev function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_function&oldid=33827
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article