# §25.16 Mathematical Applications

## §25.16(i) Distribution of Primes

In studying the distribution of primes $p\leq x$, Chebyshev (1851) introduced a function $\mathop{\psi\/}\nolimits\!\left(x\right)$ (not to be confused with the digamma function used elsewhere in this chapter), given by

 25.16.1 $\mathop{\psi\/}\nolimits\!\left(x\right)=\sum_{m=1}^{\infty}\sum_{p^{m}\leq x}% \mathop{\ln\/}\nolimits p,$ Defines: $\mathop{\psi\/}\nolimits\!\left(\NVar{x}\right)$: Chebyshev $\mathop{\psi\/}\nolimits$-function Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $m$: nonnegative integer, $p$: prime number and $x$: real variable Permalink: http://dlmf.nist.gov/25.16.E1 Encodings: TeX, pMML, png See also: info for 25.16(i)

which is related to the Riemann zeta function by

 25.16.2 $\mathop{\psi\/}\nolimits\!\left(x\right)=x-\frac{\mathop{\zeta\/}\nolimits'\!% \left(0\right)}{\mathop{\zeta\/}\nolimits\!\left(0\right)}-\sum_{\rho}\frac{x^% {\rho}}{\rho}+\mathop{o\/}\nolimits\!\left(1\right),$ $x\to\infty$,

where the sum is taken over the nontrivial zeros $\rho$ of $\mathop{\zeta\/}\nolimits\!\left(s\right)$.

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

 25.16.3 $\mathop{\psi\/}\nolimits\!\left(x\right)=x+\mathop{o\/}\nolimits\!\left(x% \right),$ $x\to\infty$.
 25.16.4 $\mathop{\psi\/}\nolimits\!\left(x\right)=x+\mathop{O\/}\nolimits\!\left(x^{% \frac{1}{2}+\epsilon}\right),$ $x\to\infty$,

(25.16.4) follows from Ingham (1932, p. 84).

## §25.16(ii) Euler Sums

Euler sums have the form

 25.16.5 $\mathop{H\/}\nolimits\!\left(s\right)=\sum_{n=1}^{\infty}\frac{h(n)}{n^{s}},$

where $h(n)$ is given by (25.11.33).

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

 25.16.6 $\mathop{H\/}\nolimits\!\left(s\right)=-\mathop{\zeta\/}\nolimits'\!\left(s% \right)+\EulerConstant\mathop{\zeta\/}\nolimits\!\left(s\right)+\frac{1}{2}% \mathop{\zeta\/}\nolimits\!\left(s+1\right)+\sum_{r=1}^{k}\mathop{\zeta\/}% \nolimits\!\left(1-2r\right)\mathop{\zeta\/}\nolimits\!\left(s+2r\right)+\sum_% {n=1}^{\infty}\frac{1}{n^{s}}\int_{n}^{\infty}\frac{\mathop{\widetilde{B}_{2k+% 1}\/}\nolimits\!\left(x\right)}{x^{2k+2}}dx,$
 25.16.7 $\mathop{H\/}\nolimits\!\left(s\right)=\frac{1}{2}\mathop{\zeta\/}\nolimits\!% \left(s+1\right)+\frac{\mathop{\zeta\/}\nolimits\!\left(s\right)}{s-1}-\sum_{r% =1}^{k}\binom{s+2r-2}{2r-1}\mathop{\zeta\/}\nolimits\!\left(1-2r\right)\mathop% {\zeta\/}\nolimits\!\left(s+2r\right)-\binom{s+2k}{2k+1}\sum_{n=1}^{\infty}% \frac{1}{n}\int_{n}^{\infty}\frac{\mathop{\widetilde{B}_{2k+1}\/}\nolimits\!% \left(x\right)}{x^{s+2k+1}}dx.$

For integer $s$ ($\geq 2$), $\mathop{H\/}\nolimits\!\left(s\right)$ can be evaluated in terms of the zeta function:

 25.16.8 $\displaystyle\mathop{H\/}\nolimits\!\left(2\right)$ $\displaystyle=2\mathop{\zeta\/}\nolimits\!\left(3\right),$ $\displaystyle\mathop{H\/}\nolimits\!\left(3\right)$ $\displaystyle=\tfrac{5}{4}\mathop{\zeta\/}\nolimits\!\left(4\right),$ Symbols: $\mathop{H\/}\nolimits\!\left(\NVar{s}\right)$: Euler sums and $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function Permalink: http://dlmf.nist.gov/25.16.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: info for 25.16(ii)
 25.16.9 $\mathop{H\/}\nolimits\!\left(a\right)=\frac{a+2}{2}\mathop{\zeta\/}\nolimits\!% \left(a+1\right)-\frac{1}{2}\sum_{r=1}^{a-2}\mathop{\zeta\/}\nolimits\!\left(r% +1\right)\mathop{\zeta\/}\nolimits\!\left(a-r\right),$ $a=2,3,4,\dots$.

Also,

 25.16.10 $\mathop{H\/}\nolimits\!\left(-2a\right)=\frac{1}{2}\mathop{\zeta\/}\nolimits\!% \left(1-2a\right)=-\frac{\mathop{B_{2a}\/}\nolimits}{4a},$ $a=1,2,3,\dots$.

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

$\mathop{H\/}\nolimits\!\left(s\right)$ is the special case $\mathop{H\/}\nolimits\!\left(s,1\right)$ of the function

 25.16.11 $\mathop{H\/}\nolimits\!\left(s,z\right)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}\sum% _{m=1}^{n}\frac{1}{m^{z}},$ $\realpart{(s+z)}>1$,

which satisfies the reciprocity law

 25.16.12 $\mathop{H\/}\nolimits\!\left(s,z\right)+\mathop{H\/}\nolimits\!\left(z,s\right% )=\mathop{\zeta\/}\nolimits\!\left(s\right)\mathop{\zeta\/}\nolimits\!\left(z% \right)+\mathop{\zeta\/}\nolimits\!\left(s+z\right),$

when both $\mathop{H\/}\nolimits\!\left(s,z\right)$ and $\mathop{H\/}\nolimits\!\left(z,s\right)$ are finite.

For further properties of $\mathop{H\/}\nolimits\!\left(s,z\right)$ see Apostol and Vu (1984). Related results are:

 25.16.13 $\displaystyle\sum_{n=1}^{\infty}\left(\frac{h(n)}{n}\right)^{2}$ $\displaystyle=\frac{17}{4}\mathop{\zeta\/}\nolimits\!\left(4\right),$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $n$: nonnegative integer and $h(n)$: sum Permalink: http://dlmf.nist.gov/25.16.E13 Encodings: TeX, pMML, png See also: info for 25.16(ii) 25.16.14 $\displaystyle\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)}$ $\displaystyle=\frac{5}{4}\mathop{\zeta\/}\nolimits\!\left(3\right),$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function and $k$: nonnegative integer Permalink: http://dlmf.nist.gov/25.16.E14 Encodings: TeX, pMML, png See also: info for 25.16(ii) 25.16.15 $\displaystyle\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{r^{2}(r+k)}$ $\displaystyle=\frac{3}{4}\mathop{\zeta\/}\nolimits\!\left(3\right).$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function and $k$: nonnegative integer Permalink: http://dlmf.nist.gov/25.16.E15 Encodings: TeX, pMML, png See also: info for 25.16(ii)

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