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1: 25.13 Periodic Zeta Function

§25.13 Periodic Zeta Function

►The notation $F(x,s)$ is used for the polylogarithm ${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}x}\right)$ with $x$ real: ►

25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

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Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

… ►Also, ►

25.13.2 $$F(x,s)=\frac{\mathrm{\Gamma }\left(1-s\right)}{{(2\pi )}^{1-s}}\left({\mathrm{e}}^{\pi \mathrm{i}(1-s)/2}\zeta (1-s,x)+{\mathrm{e}}^{\pi \mathrm{i}(s-1)/2}\zeta (1-s,1-x)\right),$$ , $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, Exercise 2, p. 273)

Permalink:

http://dlmf.nist.gov/25.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

…

2: 25.1 Special Notation

… ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

3: 23.2 Definitions and Periodic Properties

… ►The function $\zeta \left(z\right)$ is quasi-periodic: for $j=1,2,3$, …

4: 25.16 Mathematical Applications

… ►

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:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

►

25.16.7 $$H\left(s\right)=\frac{1}{2}\zeta \left(s+1\right)+\frac{\zeta \left(s\right)}{s-1}-\sum _{r=1}^k\left(\genfrac{}{}{0.0pt}{}{s+2r-2}{2r-1}\right)\zeta \left(1-2r\right)\zeta \left(s+2r\right)-\left(\genfrac{}{}{0.0pt}{}{s+2k}{2k+1}\right)\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{s+2k+1}}dx.$$

ⓘ

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:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

…

5: 25.11 Hurwitz Zeta Function

… ►

25.11.6 $$\zeta (s,a)=\frac{1}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $s\ne 1$, $\mathrm{\Re }s>-1$, $a>0$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Apostol (1985a, (25), p. 231)

Referenced by:

Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20)

Permalink:

http://dlmf.nist.gov/25.11.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.12):

Originally the integrand was incorrect because its numerator contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$.

Reported 2016-05-08 by Clemens Heuberger

See also:

Annotations for §25.11(iii), §25.11 and Ch.25

►

25.11.7 $$\zeta (s,a)=\frac{1}{a^s}+\frac{1}{{(1+a)}^s}\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}\frac{1}{{(1+a)}^{s+2k-1}}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{{(x+a)}^{s+2n+1}}dx,$$ $s\ne 1$, $a>0$, $n=1,2,3,\mathrm{\dots }$, $\mathrm{\Re }s>-2n$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz 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, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Proof sketch:

Derivable from (25.11.5) by setting $N=1$ and repeatedly integrating by parts using (1.2.6), (24.2.2), (24.2.4), (24.2.11), (24.2.12), (24.4.34).

Permalink:

http://dlmf.nist.gov/25.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(iii), §25.11 and Ch.25

… ►

25.11.19 $${\zeta }^{\prime }(s,a)=-\frac{\ln a}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{{(s-1)}^2}+\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2)\ln \left(x+a\right)}{{(x+a)}^{s+2}}dx-\frac{(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $k$: integer

Keywords:

derivative

Source:

Apostol (1985a, fourth equation, p. 231); with $k=1$, ${\phi }_2(x)=\frac{1}{2}({\tilde{B}}_2\left(x\right)-B_2)$

Referenced by:

Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20)

Permalink:

http://dlmf.nist.gov/25.11.E19

Encodings:

pMML, png, TeX

Errata (effective with 1.0.12):

Originally both integrands were incorrect because their numerators contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$.

Reported 2016-06-27 by Gergő Nemes

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

►

25.11.20 $${(-1)}^k{\zeta }^{(k)}(s,a)=\frac{{(\ln a)}^k}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum _{r=0}^{k-1}\frac{{(\ln a)}^r}{r!{(s-1)}^{k-r+1}}-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^k}{{(x+a)}^{s+2}}dx+\frac{k(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^{k-1}}{{(x+a)}^{s+2}}dx-\frac{k(k-1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^{k-2}}{{(x+a)}^{s+2}}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $k$: integer

Keywords:

derivative

Source:

Apostol (1985a, fourth equation, p. 231); with ${\phi }_2(x)=\frac{1}{2}({\tilde{B}}_2\left(x\right)-B_2)$

Referenced by:

Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20)

Permalink:

http://dlmf.nist.gov/25.11.E20

Encodings:

pMML, png, TeX

Errata (effective with 1.0.12):

Originally all three integrands were incorrect because their numerators contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$.

Reported 2016-06-27 by Gergő Nemes

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

…

6: 25.2 Definition and Expansions

… ►

25.2.9 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-\frac{1}{2}{N}^{-s}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}{N}^{1-s-2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_N^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$; $n,N=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: 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, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Proof sketch:

Derivable from (25.2.8) by repeated integration by parts.

Referenced by:

§25.18(i), (25.2.10)

Permalink:

http://dlmf.nist.gov/25.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

►

25.2.10 $$\zeta \left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$, $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: 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, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula

Proof sketch:

Derivable from (25.2.9) with $N=1$.

A&S Ref:

23.2.3

Permalink:

http://dlmf.nist.gov/25.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

…

7: 31.2 Differential Equations

… ►

§31.2(iv) Doubly-Periodic Forms

►

Jacobi’s Elliptic Form

… ►

$z={\mathrm{sn}}^2(\zeta ,k).$

… ►

31.2.8 $$\frac{d^{2}w}{{d\zeta }^2}+\left((2\gamma -1)\frac{\mathrm{cn}\zeta \mathrm{dn}\zeta }{\mathrm{sn}\zeta }-(2\delta -1)\frac{\mathrm{sn}\zeta \mathrm{dn}\zeta }{\mathrm{cn}\zeta }-(2ϵ-1)k^2\frac{\mathrm{sn}\zeta \mathrm{cn}\zeta }{\mathrm{dn}\zeta }\right)\frac{dw}{d\zeta }+4k^2(\alpha \beta {\mathrm{sn}}^2\zeta -q)w=0.$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\zeta$: change of variable and $k$: modulus

Referenced by:

§31.7(ii)

Permalink:

http://dlmf.nist.gov/31.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

►

Weierstrass’s Form

…

8: Bibliography K

… ►

The Riemann Zeta-Function. de Gruyter Expositions in Mathematics, Vol. 5, Walter de Gruyter & Co., Berlin.

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Notes:

Translated from the Russian by Neal Koblitz

External Links:

ISBN 3-11-013170-6, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

Ch.25.

Encodings:

BibTeX

… ►

Asymptotic expansions of certain $q$-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.

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External Links:

ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§17.2(i).

Encodings:

BibTeX

… ►

The Riemann Zeta-Function and Quantum Chaology. In Quantum Chaos (Varenna, 1991), Proc. Internat. School of Phys. Enrico Fermi, CXIX, pp. 145–185.

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External Links:

MathReview (Dieter H. Mayer)

Referenced by:

§25.17.

Encodings:

BibTeX

… ►

Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.13(iii), §8.22(ii).

Encodings:

BibTeX

►

Complex zeros of two incomplete Riemann zeta functions. Math. Comp. 26 (118), pp. 551–565.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.22(ii).

Encodings:

BibTeX

…

9: Errata

… ►

Originally all six integrands in these equations were incorrect because their numerators contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$. The new equations are:

25.11.6 $$\zeta (s,a)=\frac{1}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $s\ne 1$, $\mathrm{\Re }s>-1$, $a>0$

Reported 2016-05-08 by Clemens Heuberger.

25.11.19 $${\zeta }^{\prime }(s,a)=-\frac{\ln a}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{{(s-1)}^2}+\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2)\ln \left(x+a\right)}{{(x+a)}^{s+2}}dx-\frac{(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$

Reported 2016-06-27 by Gergő Nemes.

25.11.20 $${(-1)}^k{\zeta }^{(k)}(s,a)=\frac{{(\ln a)}^k}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum _{r=0}^{k-1}\frac{{(\ln a)}^r}{r!{(s-1)}^{k-r+1}}-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^k}{{(x+a)}^{s+2}}dx+\frac{k(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^{k-1}}{{(x+a)}^{s+2}}dx-\frac{k(k-1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^{k-2}}{{(x+a)}^{s+2}}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$

Reported 2016-06-27 by Gergő Nemes.

…

10: 24.17 Mathematical Applications

… ►

§24.17(ii) Spline Functions

… ►The functions … ►

Bernoulli Monosplines

… ►

§24.17(iii) Number Theory

►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …