§25.16 Mathematical Applications

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Keywords:: Riemann zeta function, applications, mathematical
Notes:: See Apostol (1976, Chapter 4), Apostol (2000), Apostol and Vu (1984), and Flajolet and Salvy (1998).
Referenced by:: §25.1
Permalink:: http://dlmf.nist.gov/25.16
See also:: Annotations for Ch.25

§25.16(i) Distribution of Primes
§25.16(ii) Euler Sums

§25.16(i) Distribution of Primes

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Keywords:: Chebyshev $\psi$ -function, Riemann hypothesis, distribution, equivalent statement, equivalent statements, prime number theorem, prime numbers
Notes:: See Apostol (1976, Chapter 4) and Apostol (2000).
Permalink:: http://dlmf.nist.gov/25.16.i
See also:: Annotations for §25.16 and Ch.25

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 $\psi\left(x\right)=\sum_{m=1}^{\infty}\sum_{p^{m}\leq x}\ln p,$

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Defines:: $\psi\left(\NVar{x}\right)$ : Chebyshev $\psi$ -function
Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer, : prime number and : 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 $\psi\left(x\right)=x-\frac{\zeta'\left(0\right)}{\zeta\left(0\right)}-\sum_{% \rho}\frac{x^{\rho}}{\rho}+o\left(1\right),$ $x\to\infty$ ,

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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 : 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 $\psi\left(x\right)=x+o\left(x\right),$ $x\to\infty$ .

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Symbols:: $\psi\left(\NVar{x}\right)$ : Chebyshev $\psi$ -function, $\sim$ : asymptotic equality, $o\left(\NVar{x}\right)$ : order less than and : 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 $\psi\left(x\right)=x+O\left(x^{\frac{1}{2}+\epsilon}\right),$ $x\to\infty$ ,

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\psi\left(\NVar{x}\right)$ : Chebyshev $\psi$ -function and : 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

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Defines:: $H\left(\NVar{s}\right)$ : Euler sums and $H\left(\NVar{s},\NVar{z}\right)$ : generalized Euler sums
Keywords:: Euler sums, Riemann zeta function, reciprocity law
Notes:: See Apostol and Vu (1984) and Flajolet and Salvy (1998).
Permalink:: http://dlmf.nist.gov/25.16.ii
See also:: Annotations for §25.16 and Ch.25

Euler sums have the form

25.16.5 $H\left(s\right)=\sum_{n=1}^{\infty}\frac{H_{n}}{n^{s}},$

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Symbols:: $H\left(\NVar{s}\right)$ : Euler sums, $H_{\NVar{n}}$ : harmonic number, : nonnegative integer, : complex variable and : 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 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 by use of the relations

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

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Symbols:: $\gamma$ : Euler’s constant, $H\left(\NVar{s}\right)$ : Euler sums, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, : nonnegative integer, : nonnegative integer, : real variable and : 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 $H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+\frac{\zeta\left(s\right)}{s-% 1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\zeta\left(1-2r\right)% \zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}% \frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{s+2k+% 1}}\,\mathrm{d}x.$

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Symbols:: $H\left(\NVar{s}\right)$ : Euler sums, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, : nonnegative integer, : nonnegative integer, : real variable and : 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 ( $\geq 2$ ), $H\left(s\right)$ can be evaluated in terms of the zeta function:

25.16.8

$H\left(2\right)=2\zeta\left(3\right),$

$H\left(3\right)=\tfrac{5}{4}\zeta\left(4\right),$

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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 $H\left(a\right)=\frac{a+2}{2}\zeta\left(a+1\right)-\frac{1}{2}\sum_{r=1}^{a-2}% \zeta\left(r+1\right)\zeta\left(a-r\right),$ $a=2,3,4,\dots$ .

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Symbols:: $H\left(\NVar{s}\right)$ : Euler sums, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and : 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 $H\left(-2a\right)=\frac{1}{2}\zeta\left(1-2a\right)=-\frac{B_{2a}}{4a},$ $a=1,2,3,\dots$ .

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $H\left(\NVar{s}\right)$ : Euler sums, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and : 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 , $r=1,2,3,\dots$ .

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

25.16.11 $H\left(s,z\right)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}\sum_{m=1}^{n}\frac{1}{m^{% z}},$ $\Re\left(s+z\right)>1$ ,

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Symbols:: $H\left(\NVar{s},\NVar{z}\right)$ : generalized Euler sums, $\Re$ : real part, : nonnegative integer, : nonnegative integer, : complex variable and : 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 $H\left(s,z\right)+H\left(z,s\right)=\zeta\left(s\right)\zeta\left(z\right)+% \zeta\left(s+z\right),$

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $H\left(\NVar{s},\NVar{z}\right)$ : generalized Euler sums, : complex variable and : 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 $\sum_{n=1}^{\infty}\left(\frac{H_{n}}{n}\right)^{2}=\frac{17}{4}\zeta\left(4% \right),$

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $H_{\NVar{n}}$ : harmonic number, : nonnegative integer and : 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 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 $\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)}=\frac{5}{4}\zeta\left(3% \right),$

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and : 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 $\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{r^{2}(r+k)}=\frac{3}{4}\zeta\left(3% \right).$

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and : 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).

25.15 Dirichlet

-functions 25.17 Physical Applications

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§25.16 Mathematical Applications

Contents

§25.16(i) Distribution of Primes

§25.16(ii) Euler Sums