§25.16 Mathematical Applications

In studying the distribution of primes $p\le x$, Chebyshev (1851) introduced a function $\psi \left(x\right)$ (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 $\rho$ of $\zeta \left(s\right)$.

The prime number theorem (27.2.3) is equivalent to the statement

The Riemann hypothesis is equivalent to the statement

for every $ϵ>0$.

Euler sums have the form

where $H_n$ is given by (25.11.33).

$H\left(s\right)$ is analytic for $\mathrm{\Re }s>1$, and can be extended meromorphically into the half-plane $\mathrm{\Re }s>-2k$ for every positive integer $k$ by use of the relations

For integer $s$ ($\ge 2$), $H\left(s\right)$ can be evaluated in terms of the zeta function:

Also,

$H\left(s\right)$ has a simple pole with residue $\zeta \left(1-2r\right)$ ($=-B_{2r}/(2r)$) at each odd negative integer $s=1-2r$, $r=1,2,3,\mathrm{\dots }$.

$H\left(s\right)$ is the special case $H(s,1)$ of the function

which satisfies the reciprocity law

when both $H(s,z)$ and $H(z,s)$ are finite.

For further properties of $H(s,z)$ see Apostol and Vu (1984). Related results are:

For further generalizations, see Flajolet and Salvy (1998).