Riemann zeta function

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1: 25.1 Special Notation

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$k,m,n$	nonnegative integers.
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►The main function treated in this chapter is the Riemann zeta function $\zeta \left(s\right)$. … ►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 $L(s,\chi )$.

2: 8.22 Mathematical Applications

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§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function

… ►If ${\zeta }_x(s)$ denotes the incomplete Riemann zeta function defined by …so that ${lim}_{x\to \mathrm{\infty }}{\zeta }_x(s)=\zeta \left(s\right)$, then ►

8.22.3 $${\zeta }_x(s)=\sum _{k=1}^{\mathrm{\infty }}{k}^{-s}P(s,kx),$$ $\mathrm{\Re }s>1$.

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Symbols:

$P(a,z)$: normalized incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $k$: nonnegative integer and ${\zeta }_x(s)$: incomplete Riemann zeta function

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http://dlmf.nist.gov/8.22.E3

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Annotations for §8.22(ii), §8.22 and Ch.8

►For further information on ${\zeta }_x(s)$, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …

3: 25.7 Integrals

§25.7 Integrals

►For definite integrals of the Riemann zeta function see Prudnikov et al. (1986b, §2.4), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).

4: 25.17 Physical Applications

§25.17 Physical Applications

… ►See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). …

5: 25.20 Approximations

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Cody et al. (1971) gives rational approximations for $\zeta \left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\le s\le 5$, $5\le s\le 11$, $11\le s\le 25$, $25\le s\le 55$. Precision is varied, with a maximum of 20S.

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Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta \left(s+1\right)$ and $\zeta \left(s+k\right)$, $k=2,3,4,5,8$, for $0\le s\le 1$ (23D).

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Antia (1993) gives minimax rational approximations for $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals and , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$. For each $s$ there are three sets of approximations, with relative maximum errors ${10}^{-4},{10}^{-8},{10}^{-12}$.

6: 25.3 Graphics

§25.3 Graphics

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Figure 25.3.1: Riemann zeta function $\zeta \left(x\right)$ and its derivative ${\zeta }^{\prime }\left(x\right)$, $-20\le x\le 10$. Magnify

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Figure 25.3.2: Riemann zeta function $\zeta \left(x\right)$ and its derivative ${\zeta }^{\prime }\left(x\right)$, $-12\le x\le -2$. Magnify

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Figure 25.3.3: Modulus of the Riemann zeta function $|\zeta \left(x+\mathrm{i}y\right)|$, $-4\le x\le 4$, $-10\le y\le 40$. Magnify 3D Help

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Figure 25.3.6: $Z(t)$, $10000\le t\le 10050$. Magnify

7: 25.18 Methods of Computation

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§25.18(i) Function Values and Derivatives

… ►Calculations relating to derivatives of $\zeta \left(s\right)$ and/or $\zeta (s,a)$ can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988). … ►

§25.18(ii) Zeros

►Most numerical calculations of the Riemann zeta function are concerned with locating zeros of $\zeta \left(\frac{1}{2}+\mathrm{i}t\right)$ in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of $\zeta \left(s\right)$ lie on the critical line $\mathrm{\Re }s=\frac{1}{2}$. …

8: 27.4 Euler Products and Dirichlet Series

… ►The completely multiplicative function $f(n)=n^{-s}$ gives the Euler product representation of the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)): … ►The Riemann zeta function is the prototype of series of the form … ►

27.4.5 $$\sum _{n=1}^{\mathrm{\infty }}\mu \left(n\right)n^{-s}=\frac{1}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mu \left(n\right)$: Möbius function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

A&S Ref:

24.3.1 I.B

Permalink:

http://dlmf.nist.gov/27.4.E5

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Annotations for §27.4 and Ch.27

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27.4.7 $$\sum _{n=1}^{\mathrm{\infty }}\lambda \left(n\right)n^{-s}=\frac{\zeta \left(2s\right)}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.4.E7

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Annotations for §27.4 and Ch.27

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27.4.12 $$\sum _{n=1}^{\mathrm{\infty }}\mathrm{\Lambda }\left(n\right)n^{-s}=-\frac{{\zeta }^{\prime }\left(s\right)}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Lambda }\left(n\right)$: Mangoldt’s function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

Referenced by:

§27.4

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http://dlmf.nist.gov/27.4.E12

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Annotations for §27.4 and Ch.27

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9: 25.4 Reflection Formulas

§25.4 Reflection Formulas

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25.4.1 $$\zeta \left(1-s\right)=2{(2\pi )}^{-s}\cos \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(s\right)\zeta \left(s\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $s$: complex variable

Keywords:

reflection formula

Source:

Apostol (1976, (12), p. 259)

Referenced by:

§25.10(i), §5.17

Permalink:

http://dlmf.nist.gov/25.4.E1

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Annotations for §25.4 and Ch.25

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25.4.2 $$\zeta \left(s\right)=2{(2\pi )}^{s-1}\sin \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(1-s\right)\zeta \left(1-s\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $s$: complex variable

Keywords:

reflection formula

Source:

Apostol (1976, (13), p. 259)

A&S Ref:

23.2.6

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http://dlmf.nist.gov/25.4.E2

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Annotations for §25.4 and Ch.25

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25.4.4 $$\xi \left(s\right)=\frac{1}{2}s(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s\right){\pi }^{-s/2}\zeta \left(s\right).$$

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Defines:

$\xi \left(s\right)$: Riemann’s $\xi$-function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $s$: complex variable

Keywords:

definition

Source:

Apostol (1976, p. 260)

Permalink:

http://dlmf.nist.gov/25.4.E4

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Annotations for §25.4 and Ch.25

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25.4.5 $${(-1)}^k{\zeta }^{(k)}\left(1-s\right)=\frac{2}{{(2\pi )}^s}\sum _{m=0}^k\sum _{r=0}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\left(\mathrm{\Re }\left(c^{k-m}\right)\cos \left(\frac{1}{2}\pi s\right)+\mathrm{\Im }\left(c^{k-m}\right)\sin \left(\frac{1}{2}\pi s\right)\right){\mathrm{\Gamma }}^{(r)}\left(s\right){\zeta }^{(m-r)}\left(s\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part, $\sin z$: sine function, $k$: nonnegative integer, $m$: nonnegative integer, $s$: complex variable and $c$

Keywords:

derivative

Source:

Apostol (1985a, (6), p. 224)

Permalink:

http://dlmf.nist.gov/25.4.E5

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Annotations for §25.4 and Ch.25

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10: 25.8 Sums

§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

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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)

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

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Annotations for §25.8 and Ch.25

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25.8.4 $$\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^k}{k}(\zeta \left(nk\right)-1)=\ln \left(\prod _{j=0}^{n-1}\mathrm{\Gamma }\left(2-{\mathrm{e}}^{(2j+1)\pi \mathrm{i}/n}\right)\right),$$ $n=2,3,4,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $n$: nonnegative integer

Keywords:

infinite series

Source:

Adamchik and Srivastava (1998, (2.7), p. 135)

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http://dlmf.nist.gov/25.8.E4

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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

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Annotations for §25.8 and Ch.25

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