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: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

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§25.11(i) Definition

… ►The Riemann zeta function is a special case: … ►

§25.11(ii) Graphics

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Figure 25.11.2: Hurwitz zeta function $\zeta (x,a)$, $-19.5\le x\le 10$, $0.02\le a\le 1$. Magnify 3D Help

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3: 22.16 Related Functions

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§22.16(iii) Jacobi’s Zeta Function

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Definition

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Properties

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Figure 22.16.3: Jacobi’s zeta function $\mathrm{Z}\left(x|k\right)$ for $0\le x\le 10\pi$ and $k=0.4,0.7,0.99,0.999999$. Magnify

4: 8.22 Mathematical Applications

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

… ►See Paris and Cang (1997). ►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 …For further information on ${\zeta }_x(s)$, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …

5: 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).

6: 25.17 Physical Applications

§25.17 Physical Applications

… ►See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). ►The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). …It has been found possible to perform such regularizations by equating the divergent sums to zeta functions and associated functions (Elizalde (1995)).

7: 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: … ►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

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25.13.3 $$\zeta (1-s,x)=\frac{\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}F(x,s)+{\mathrm{e}}^{\pi \mathrm{i}s/2}F(-x,s)\right),$$ $\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$.

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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, (10), p. 257)

Notes:

This formula is equivalent to (25.11.9).

Referenced by:

Erratum (V1.1.4) for Equation (25.13.3)

Permalink:

http://dlmf.nist.gov/25.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

8: 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). ►For the Hurwitz zeta function $\zeta (s,a)$ see Spanier and Oldham (1987, p. 653) and Coffey (2009). … ►

§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 $\mathrm{\Re }s=\frac{1}{2}$. …

9: 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 ${10}^{-4},{10}^{-8},{10}^{-12}$.

10: 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