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1: 4.11 Sums

… ►For infinite series involving logarithms and/or exponentials, see Gradshteyn and Ryzhik (2015, Chapter 1), Hansen (1975, §44), and Prudnikov et al. (1986a, Chapter 5).

2: 4.22 Infinite Products and Partial Fractions

§4.22 Infinite Products and Partial Fractions

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3: 4.36 Infinite Products and Partial Fractions

§4.36 Infinite Products and Partial Fractions

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4: 25.8 Sums

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25.8.1 $$\sum _{k=2}^{\mathrm{\infty }}\left(\zeta \left(k\right)-1\right)=1.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function and $k$: nonnegative integer

Keywords:

infinite series

Source:

Adamchik and Srivastava (1998, (1.4), p. 132)

Permalink:

http://dlmf.nist.gov/25.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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25.8.2 $$\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(s+k\right)}{(k+1)!}\left(\zeta \left(s+k\right)-1\right)=\mathrm{\Gamma }\left(s-1\right),$$ $s\ne 1,0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Source:

Landau (1953, (2), p. 273); with (5.2.4), (5.2.5)

Permalink:

http://dlmf.nist.gov/25.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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25.8.5 $$\sum _{k=2}^{\mathrm{\infty }}\zeta \left(k\right)z^k=-\gamma z-z\psi \left(1-z\right),$$ .

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

Keywords:

infinite series

Source:

Erdélyi et al. (1953a, (1.17.5), p. 45); with $z\mapsto -z$

Referenced by:

(25.8.7)

Permalink:

http://dlmf.nist.gov/25.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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25.8.9 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac{1}{2}\ln 2.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function and $k$: nonnegative integer

Keywords:

infinite series

Source:

Srivastava and Choi (2001, (467), p. 212)

Permalink:

http://dlmf.nist.gov/25.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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25.8.10 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}-\frac{7}{4{\pi }^2}\zeta \left(3\right).$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $k$: nonnegative integer

Keywords:

infinite series

Source:

Ewell (1990, (1), p. 219)

Permalink:

http://dlmf.nist.gov/25.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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5: 25.2 Definition and Expansions

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25.2.1 $$\zeta \left(s\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}.$$

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Defines:

$\zeta \left(s\right)$: Riemann zeta function

Symbols:

$n$: nonnegative integer and $s$: complex variable

Keywords:

definition, infinite series

Sources:

Apostol (1976, Chapter 12); Riemann (1859, p. 136)

A&S Ref:

23.2.1

Referenced by:

(25.11.2), (25.2.2), (25.2.4), §25.2(ii), Erratum (V1.0.17) for Equation (25.2.4)

Permalink:

http://dlmf.nist.gov/25.2.E1

Encodings:

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See also:

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

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§25.2(ii) Other Infinite Series

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25.2.4 $$\zeta \left(s\right)=\frac{1}{s-1}+\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{n!}{\gamma }_n{(s-1)}^n,$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series, Laurent series

Sources:

Hardy (1912, p. 216); mistakenly missing factor of ${(-1)}^n/n!$; Ivić (1985, (1.11), p. 4)

A&S Ref:

23.2.5 (with unnecessary constraint removed)

Referenced by:

Erratum (V1.0.17) for Equation (25.2.4), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E4

Encodings:

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Clarification (effective with 1.0.17):

Originally this equation had the constraint $\mathrm{\Re }s>0$. This constraint is unnecessary because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$.

Suggested 2017-12-05 by John Harper

See also:

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

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§25.2(iv) Infinite Products

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25.2.11 $$\zeta \left(s\right)=\prod _p{(1-p^{-s})}^{-1},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $p$: prime number and $s$: complex variable

Keywords:

infinite product, primes

Source:

Apostol (1976, p. 231)

A&S Ref:

23.2.2

Referenced by:

§25.10(i)

Permalink:

http://dlmf.nist.gov/25.2.E11

Encodings:

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See also:

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

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6: Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►This pattern is analogous to one that would be seen in fluid flow generated by a semi-infinite line of vortices. …

7: 5.8 Infinite Products

§5.8 Infinite Products

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8: 1.9 Calculus of a Complex Variable

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§1.9(v) Infinite Sequences and Series

… ►This sequence converges pointwise to a function $f(z)$ if … ► … ►

§1.9(vii) Inversion of Limits

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Dominated Convergence Theorem

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9: 24.8 Series Expansions

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§24.8(i) Fourier Series

… ►If $n=1,2,\mathrm{\dots }$ and $0\le x\le 1$, then … ►

§24.8(ii) Other Series

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24.8.9 $$E_{2n}={(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n}}{\cosh \left(\frac{1}{2}\pi k\right)}-4\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(2k+1)}^{2n}}{{\mathrm{e}}^{2\pi (2k+1)}-1},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $k$: integer and $n$: integer

Keywords:

Euler polynomials, infinite series expansions, other

Permalink:

http://dlmf.nist.gov/24.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.8(ii), §24.8 and Ch.24

10: 1.10 Functions of a Complex Variable

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§1.10(ix) Infinite Products

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Weierstrass Product

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§1.10(x) Infinite Partial Fractions

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Mittag-Leffler’s Expansion

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