Main Index
Number Theory
Arithmetic functions
Main Index
Number Theory
Sequences
Primes
Subject Index
comment on the page
There are two different functions which are known as the Chebyshev’s functions:
The first Chebyshev function
![]() | (1) |
where the summation is over all positive primes
, and
the second Chebyshev’s function
![]() | (2) |
where the summation runs over the all prime powers with
(
is the von Mangoldt’s function)
.
Clearly, the first function is nothing else as the logarithm of the product of all primes , while the second one is the logarithm of the least common multiple of all positive integers
. We therefore have
![]() | (3) |
Theorem. For we have
![]() | (4) |
Corollary.
![]() | (5) |
Theorem. If we have
![]() | (6) |
![]() | (7) |
Theorem. The following relations are equivalent
![]() | (8) |
![]() | (9) |
![]() | (10) |
Cite this web-page as:
Štefan Porubský: Chebyshev’s functions.