§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

25.16.1		$$\psi \left(x\right)=\sum _{m=1}^{\mathrm{\infty }}\sum _{p^m\le x}\ln p,$$
ⓘ Defines: $\psi \left(x\right)$: Chebyshev $\psi$-function Symbols: $\ln z$: 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		$$\psi \left(x\right)=x-\frac{{\zeta }^{\prime }\left(0\right)}{\zeta \left(0\right)}-\sum _{\rho }\frac{x^{\rho }}{\rho }+o\left(1\right),$$
$x\to \mathrm{\infty }$,
ⓘ Symbols: $\psi \left(x\right)$: Chebyshev $\psi$-function, $\zeta \left(s\right)$: Riemann zeta function, $o\left(x\right)$: order less than and $x$: real variable Keywords: asymptotic approximation Source: Apostol (2000, p. 9) Referenced by: §25.16(i) 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		$$\psi \left(x\right)=x+o\left(x\right),$$
$x\to \mathrm{\infty }$.
ⓘ Symbols: $\psi \left(x\right)$: Chebyshev $\psi$-function, $\sim$: asymptotic equality, $o\left(x\right)$: 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 $\psi \left(x\right)=x(1+o\left(1\right))$ 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		$$\psi \left(x\right)=x+O\left(x^{\frac{1}{2}+ϵ}\right),$$
$x\to \mathrm{\infty }$,
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\psi \left(x\right)$: Chebyshev $\psi$-function and $x$: real variable Keywords: asymptotic approximation Source: Apostol (2000, p. 9) Referenced by: §25.16(i) 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 $ϵ>0$.

Euler sums have the form

25.16.5		$$H\left(s\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{h(n)}{n^s},$$
ⓘ Symbols: $H\left(s\right)$: Euler sums, $n$: nonnegative integer, $s$: complex variable and $h(n)$: sum Keywords: Euler sum, infinite series, series representation Source: Apostol and Vu (1984, p. 85) Permalink: http://dlmf.nist.gov/25.16.E5 Encodings: TeX, pMML, png 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 $\mathrm{\Re }s>1$, and can be extended meromorphically into the half-plane $\mathrm{\Re }s>-2k$ for every positive integer $k$ by use of the relations

25.16.6		$$H\left(s\right)=-{\zeta }^{\prime }\left(s\right)+\gamma \zeta \left(s\right)+\frac{1}{2}\zeta \left(s+1\right)+\sum _{r=1}^k\zeta \left(1-2r\right)\zeta \left(s+2r\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{2k+2}}dx,$$
ⓘ Symbols: $\gamma$: Euler’s constant, $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$, $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable Keywords: Euler sum, finite sum, improper integral, infinite series, integral representation, series representation Source: Apostol and Vu (1984, (3), p. 87) Permalink: http://dlmf.nist.gov/25.16.E6 Encodings: TeX, pMML, png See also: Annotations for §25.16(ii), §25.16 and Ch.25

25.16.7		$$H\left(s\right)=\begin{array}{l}\frac{1}{2}\zeta \left(s+1\right)+\frac{\zeta \left(s\right)}{s-1}-\sum _{r=1}^k\left(\genfrac{}{}{0.0pt}{}{s+2r-2}{2r-1}\right)\zeta \left(1-2r\right)\zeta \left(s+2r\right)\\ -\left(\genfrac{}{}{0.0pt}{}{s+2k}{2k+1}\right)\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{s+2k+1}}dx.\end{array}$$
ⓘ Symbols: $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$, $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable Keywords: Euler sum, finite sum, improper integral, infinite series, integral representation, series representation Source: Apostol and Vu (1984, (6), p. 88) Permalink: http://dlmf.nist.gov/25.16.E7 Encodings: TeX, pMML, png See also: Annotations for §25.16(ii), §25.16 and Ch.25

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

25.16.8		$H\left(2\right)$	$=2\zeta \left(3\right),$
		$H\left(3\right)$	$=\frac{5}{4}\zeta \left(4\right),$
ⓘ Symbols: $H\left(s\right)$: Euler sums and $\zeta \left(s\right)$: 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		$$H\left(a\right)=\frac{a+2}{2}\zeta \left(a+1\right)-\frac{1}{2}\sum _{r=1}^{a-2}\zeta \left(r+1\right)\zeta \left(a-r\right),$$
$a=2,3,4,\mathrm{\dots }$.
ⓘ Symbols: $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function and $a$: real or complex parameter Source: Basu and Apostol (2000, (6.4), p. 416) 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		$$H\left(-2a\right)=\frac{1}{2}\zeta \left(1-2a\right)=-\frac{B_{2a}}{4a},$$
$a=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: 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 $s=1-2r$, $r=1,2,3,\mathrm{\dots }$.

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

25.16.11		$$H(s,z)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}\sum _{m=1}^n\frac{1}{m^z},$$
$\mathrm{\Re }\left(s+z\right)>1$,
ⓘ Symbols: $H(s,z)$: generalized Euler sums, $\mathrm{\Re }$: 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, (11), p. 90) 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		$$H(s,z)+H(z,s)=\zeta \left(s\right)\zeta \left(z\right)+\zeta \left(s+z\right),$$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $H(s,z)$: 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 $H(s,z)$ and $H(z,s)$ are finite.

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

25.16.13		${\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\left({\displaystyle \frac{h(n)}{n}}\right)}^2$	$={\displaystyle \frac{17}{4}}\zeta \left(4\right),$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $n$: nonnegative integer and $h(n)$: sum Keywords: harmonic number, infinite series, special value Source: Flajolet and Salvy (1998, (4-1), p. 24) Permalink: http://dlmf.nist.gov/25.16.E13 Encodings: TeX, pMML, png See also: Annotations for §25.16(ii), §25.16 and Ch.25
25.16.14		${\displaystyle \sum _{r=1}^{\mathrm{\infty }}}{\displaystyle \sum _{k=1}^r}{\displaystyle \frac{1}{rk(r+k)}}$	$={\displaystyle \frac{5}{4}}\zeta \left(3\right),$
ⓘ Symbols: $\zeta \left(s\right)$: 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		${\displaystyle \sum _{r=1}^{\mathrm{\infty }}}{\displaystyle \sum _{k=1}^r}{\displaystyle \frac{1}{r^2(r+k)}}$	$={\displaystyle \frac{3}{4}}\zeta \left(3\right).$
ⓘ Symbols: $\zeta \left(s\right)$: 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).