In studying the distribution of primes , Chebyshev (1851) introduced a function (not to be confused with the digamma function used elsewhere in this chapter), given by
which is related to the Riemann zeta function by
where the sum is taken over the nontrivial zeros of .
The prime number theorem (27.2.3) is equivalent to the statement
The Riemann hypothesis is equivalent to the statement
for every .
Euler sums have the form
where is given by (25.11.33).
is analytic for , and can be extended meromorphically into the half-plane for every positive integer by use of the relations
For integer (), can be evaluated in terms of the zeta function:
has a simple pole with residue () at each odd negative integer , .
is the special case of the function
which satisfies the reciprocity law
when both and are finite.
For further properties of see Apostol and Vu (1984). Related results are:
For further generalizations, see Flajolet and Salvy (1998).