# §25.16 Mathematical Applications

## §25.16(i) Distribution of Primes

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

 25.16.1 $\psi\left(x\right)=\sum_{m=1}^{\infty}\sum_{p^{m}\leq x}\ln p,$ ⓘ Defines: $\psi\left(\NVar{x}\right)$: Chebyshev $\psi$-function Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $m$: nonnegative integer, $p$: prime number and $x$: real variable Keywords: definition, double series, infinite series, series representation Source: Apostol (1976, p. 75) Permalink: http://dlmf.nist.gov/25.16.E1 Encodings: TeX, pMML, png See also: Annotations for §25.16(i), §25.16 and Ch.25

which is related to the Riemann zeta function by

 25.16.2 $\psi\left(x\right)=x-\frac{\zeta'\left(0\right)}{\zeta\left(0\right)}-\sum_{% \rho}\frac{x^{\rho}}{\rho}+o\left(1\right),$ $x\to\infty$, ⓘ Symbols: $\psi\left(\NVar{x}\right)$: Chebyshev $\psi$-function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $o\left(\NVar{x}\right)$: order less than and $x$: real variable Keywords: asymptotic approximation Source: Apostol (2000, p. 9) Permalink: http://dlmf.nist.gov/25.16.E2 Encodings: TeX, pMML, png See also: Annotations for §25.16(i), §25.16 and Ch.25

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

 25.16.3 $\psi\left(x\right)=x+o\left(x\right),$ $x\to\infty$. ⓘ Symbols: $\psi\left(\NVar{x}\right)$: Chebyshev $\psi$-function, $\sim$: asymptotic equality, $o\left(\NVar{x}\right)$: order less than and $x$: real variable Keywords: asymptotic approximation Source: Apostol (1976, (10), p. 79); since $\psi\left(x\right)\sim x$ is equivalent to $\psi\left(x\right)=x+o\left(x\right)$ Referenced by: §25.10(i) Permalink: http://dlmf.nist.gov/25.16.E3 Encodings: TeX, pMML, png See also: Annotations for §25.16(i), §25.16 and Ch.25

The Riemann hypothesis is equivalent to the statement

 25.16.4 $\psi\left(x\right)=x+O\left(x^{\frac{1}{2}+\epsilon}\right),$ $x\to\infty$, ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\psi\left(\NVar{x}\right)$: Chebyshev $\psi$-function and $x$: real variable Keywords: asymptotic approximation Source: Apostol (2000, p. 9) Permalink: http://dlmf.nist.gov/25.16.E4 Encodings: TeX, pMML, png See also: Annotations for §25.16(i), §25.16 and Ch.25

for every $\epsilon>0$.

## §25.16(ii) Euler Sums

Euler sums have the form

 25.16.5 $H\left(s\right)=\sum_{n=1}^{\infty}\frac{H_{n}}{n^{s}},$ ⓘ Symbols: $H\left(\NVar{s}\right)$: Euler sums, $H_{\NVar{n}}$: harmonic number, $n$: nonnegative integer, $s$: complex variable and $h(n)$: harmonic number Keywords: Euler sum, infinite series, series representation Source: Apostol and Vu (1984, p. 85) Referenced by: Erratum (V1.1.4) for Notation Permalink: http://dlmf.nist.gov/25.16.E5 Encodings: TeX, pMML, png Notation (effective with 1.1.4): The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_{n}$. Suggested 2021-08-23 by Gergő Nemes See also: Annotations for §25.16(ii), §25.16 and Ch.25

where $H_{n}$ is given by (25.11.33).

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

 25.16.6 $H\left(s\right)=-\zeta'\left(s\right)+\gamma\zeta\left(s\right)+\frac{1}{2}% \zeta\left(s+1\right)+\sum_{r=1}^{k}\zeta\left(1-2r\right)\zeta\left(s+2r% \right)+\sum_{n=1}^{\infty}\frac{1}{n^{s}}\int_{n}^{\infty}\frac{\widetilde{B}% _{2k+1}\left(x\right)}{x^{2k+2}}\,\mathrm{d}x,$
 25.16.7 $H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+\frac{\zeta\left(s\right)}{s-% 1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\zeta\left(1-2r\right)% \zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}% \frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{s+2k+% 1}}\,\mathrm{d}x.$

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

 25.16.8 $\displaystyle H\left(2\right)$ $\displaystyle=2\zeta\left(3\right),$ $\displaystyle H\left(3\right)$ $\displaystyle=\tfrac{5}{4}\zeta\left(4\right),$ ⓘ Symbols: $H\left(\NVar{s}\right)$: Euler sums and $\zeta\left(\NVar{s}\right)$: Riemann zeta function Keywords: special value Source: Apostol and Vu (1984, p. 92); with (25.16.9), (25.6.1) Permalink: http://dlmf.nist.gov/25.16.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §25.16(ii), §25.16 and Ch.25
 25.16.9 $H\left(a\right)=\frac{a+2}{2}\zeta\left(a+1\right)-\frac{1}{2}\sum_{r=1}^{a-2}% \zeta\left(r+1\right)\zeta\left(a-r\right),$ $a=2,3,4,\dots$. ⓘ Symbols: $H\left(\NVar{s}\right)$: Euler sums, $\zeta\left(\NVar{s}\right)$: Riemann zeta function and $a$: real or complex parameter Source: Apostol and Vu (1984, (12), p. 92) Referenced by: (25.16.8) Permalink: http://dlmf.nist.gov/25.16.E9 Encodings: TeX, pMML, png See also: Annotations for §25.16(ii), §25.16 and Ch.25

Also,

 25.16.10 $H\left(-2a\right)=\frac{1}{2}\zeta\left(1-2a\right)=-\frac{B_{2a}}{4a},$ $a=1,2,3,\dots$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $H\left(\NVar{s}\right)$: Euler sums, $\zeta\left(\NVar{s}\right)$: Riemann zeta function and $a$: real or complex parameter Keywords: Euler sum, special value Source: Apostol and Vu (1984, (7), p. 88) Permalink: http://dlmf.nist.gov/25.16.E10 Encodings: TeX, pMML, png See also: Annotations for §25.16(ii), §25.16 and Ch.25

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

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

 25.16.11 $H\left(s,z\right)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}\sum_{m=1}^{n}\frac{1}{m^{% z}},$ $\Re\left(s+z\right)>1$, ⓘ Symbols: $H\left(\NVar{s},\NVar{z}\right)$: generalized Euler sums, $\Re$: real part, $m$: nonnegative integer, $n$: nonnegative integer, $s$: complex variable and $z$: complex variable Keywords: Euler sum, infinite series, series representation Source: Apostol and Vu (1984, (1), p. 86) Permalink: http://dlmf.nist.gov/25.16.E11 Encodings: TeX, pMML, png See also: Annotations for §25.16(ii), §25.16 and Ch.25

which satisfies the reciprocity law

 25.16.12 $H\left(s,z\right)+H\left(z,s\right)=\zeta\left(s\right)\zeta\left(z\right)+% \zeta\left(s+z\right),$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $H\left(\NVar{s},\NVar{z}\right)$: generalized Euler sums, $s$: complex variable and $z$: complex variable Keywords: addition formula, connection formula Source: Apostol and Vu (1984, (11), p. 91) Permalink: http://dlmf.nist.gov/25.16.E12 Encodings: TeX, pMML, png See also: Annotations for §25.16(ii), §25.16 and Ch.25

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

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

 25.16.13 $\displaystyle\sum_{n=1}^{\infty}\left(\frac{H_{n}}{n}\right)^{2}$ $\displaystyle=\frac{17}{4}\zeta\left(4\right),$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $H_{\NVar{n}}$: harmonic number, $n$: nonnegative integer and $h(n)$: harmonic number Keywords: harmonic number, infinite series, special value Source: Flajolet and Salvy (1998, (4-1), p. 24) Referenced by: Erratum (V1.1.4) for Notation Permalink: http://dlmf.nist.gov/25.16.E13 Encodings: TeX, pMML, png Notation (effective with 1.1.4): The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_{n}$. Suggested 2021-08-23 by Gergő Nemes See also: Annotations for §25.16(ii), §25.16 and Ch.25 25.16.14 $\displaystyle\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)}$ $\displaystyle=\frac{5}{4}\zeta\left(3\right),$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function and $k$: nonnegative integer Keywords: infinite series, special value Source: Apostol and Vu (1984, (14), p. 94) Permalink: http://dlmf.nist.gov/25.16.E14 Encodings: TeX, pMML, png See also: Annotations for §25.16(ii), §25.16 and Ch.25 25.16.15 $\displaystyle\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{r^{2}(r+k)}$ $\displaystyle=\frac{3}{4}\zeta\left(3\right).$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function and $k$: nonnegative integer Keywords: infinite series, special value Source: Apostol and Vu (1984, (15), p. 94) Permalink: http://dlmf.nist.gov/25.16.E15 Encodings: TeX, pMML, png See also: Annotations for §25.16(ii), §25.16 and Ch.25

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