functions s(ϵ,ℓ;r),c(ϵ,ℓ;r)

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11: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau \left(n\right)$, and the values can be checked by the congruence (27.14.20). …

12: 7 Error Functions, Dawson’s and Fresnel Integrals

Chapter 7 Error Functions, Dawson’s and Fresnel Integrals

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13: 20.8 Watson’s Expansions

§20.8 Watson’s Expansions

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14: 25.4 Reflection Formulas

… ►

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

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

►

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

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

… ►

25.4.3 $$\xi \left(s\right)=\xi \left(1-s\right),$$

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Symbols:

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

Keywords:

reflection formula

Source:

Apostol (1976, p. 260)

Permalink:

http://dlmf.nist.gov/25.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

►where $\xi \left(s\right)$ is Riemann’s $\xi$-function, defined by: ►

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

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

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15: 27.2 Functions

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27.2.11 $$J_k\left(n\right)=\sum _{\left(\left(d_1,\mathrm{\dots },d_k\right),n\right)=1}1,$$

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Defines:

$J_k\left(n\right)$: Jordan’s function

Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer and $n$: positive integer

Referenced by:

§27.2(i)

Permalink:

http://dlmf.nist.gov/27.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►This is Jordan’s function. … ►

27.2.13

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Defines:

$\lambda \left(n\right)$: Liouville’s function

Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer and $a$: primitive root

Permalink:

http://dlmf.nist.gov/27.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

►This is Liouville’s function. … ►This is Mangoldt’s function. …

16: 7.16 Generalized Error Functions

§7.16 Generalized Error Functions

►Generalizations of the error function and Dawson’s integral are ${\int }_0^x{\mathrm{e}}^{-t^p}dt$ and ${\int }_0^x{\mathrm{e}}^{t^p}dt$. …

17: 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 …The function $F(s)$ is a generating function, or more precisely, a Dirichlet generating function, for the coefficients. … ►

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

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

… ►

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

Permalink:

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

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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18: 8.1 Special Notation

… ►Alternative notations include: Prym’s functions $P_z(a)=\gamma (a,z)$, $Q_z(a)=\mathrm{\Gamma }(a,z)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma (a+1,z)$, $[a,z]!=\mathrm{\Gamma }(a+1,z)$, Dingle (1973); $B(a,b,x)={\mathrm{B}}_x(a,b)$, $I(a,b,x)=I_x(a,b)$, Magnus et al. (1966); $\mathrm{Si}(a,x)\to \mathrm{Si}(1-a,x)$, $\mathrm{Ci}(a,x)\to \mathrm{Ci}(1-a,x)$, Luke (1975).

19: 27.6 Divisor Sums

… ►

27.6.1

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Symbols:

$\lambda \left(n\right)$: Liouville’s function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

… ►

27.6.5 $$\sum _{d|n}\frac{|\mu \left(d\right)|}{\varphi \left(d\right)}=\frac{n}{\varphi \left(n\right)},$$

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Symbols:

$\varphi \left(n\right)$: Euler’s totient, $\mu \left(n\right)$: Möbius function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

… ►

27.6.7 $$\sum _{d|n}\mu \left(d\right){\left(\frac{n}{d}\right)}^k=J_k\left(n\right),$$

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Symbols:

$J_k\left(n\right)$: Jordan’s function, $\mu \left(n\right)$: Möbius function, $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

►

27.6.8 $$\sum _{d|n}{J}_k\left(d\right)=n^k.$$

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Symbols:

$J_k\left(n\right)$: Jordan’s function, $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

20: 15.15 Sums

… ►

15.15.1 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{c};\frac{1}{z})={\left(1-\frac{z_0}{z}\right)}^{-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s}{s!}𝐅(\genfrac{}{}{0pt}{}{-s,b}{c};\frac{1}{z_0}){\left(1-\frac{z}{z_0}\right)}^{-s}.$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.19(i)

Permalink:

http://dlmf.nist.gov/15.15.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §15.15 and Ch.15

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