Euler%20totient

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11: 27.7 Lambert Series as Generating Functions

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27.7.4 $$\sum _{n=1}^{\mathrm{\infty }}\varphi \left(n\right)\frac{x^n}{1-x^n}=\frac{x}{{(1-x)}^2},$$

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Symbols:

$\varphi \left(n\right)$: Euler’s totient, $n$: positive integer and $x$: real number

A&S Ref:

24.3.2 I.B

Permalink:

http://dlmf.nist.gov/27.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

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12: 27.12 Asymptotic Formulas: Primes

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27.12.8 $$\frac{\mathrm{li}\left(x\right)}{\varphi \left(m\right)}+O\left(x\exp \left(-\lambda (\alpha ){(\ln x)}^{1/2}\right)\right),$$ $m\le {(\ln x)}^{\alpha }$, $\alpha >0$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $m$: positive integer and $x$: real number

Referenced by:

Erratum (V1.0.17) for Equation (27.12.8)

Permalink:

http://dlmf.nist.gov/27.12.E8

Encodings:

pMML, png, TeX

Errata (effective with 1.0.17):

Originally the first term was given incorrectly by $\frac{x}{\varphi \left(m\right)}$.

Reported 2017-12-04 by Gergő Nemes

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

►where $\lambda (\alpha )$ depends only on $\alpha$, and $\varphi \left(m\right)$ is the Euler totient function (§27.2). …

13: 5.22 Tables

… ►Abramowitz and Stegun (1964, Chapter 6) tabulates $\mathrm{\Gamma }\left(x\right)$, $\ln \mathrm{\Gamma }\left(x\right)$, $\psi \left(x\right)$, and ${\psi }^{\prime }\left(x\right)$ for $x=1(.005)2$ to 10D; ${\psi }^{\prime \prime }\left(x\right)$ and ${\psi }^{(3)}\left(x\right)$ for $x=1(.01)2$ to 10D; $\mathrm{\Gamma }\left(n\right)$, $1/\mathrm{\Gamma }\left(n\right)$, $\mathrm{\Gamma }\left(n+\frac{1}{2}\right)$, $\psi \left(n\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{1}{3}\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{1}{2}\right)$, and ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{2}{3}\right)$ for $n=1(1)101$ to 8–11S; $\mathrm{\Gamma }\left(n+1\right)$ for $n=100(100)1000$ to 20S. Zhang and Jin (1996, pp. 67–69 and 72) tabulates $\mathrm{\Gamma }\left(x\right)$, $1/\mathrm{\Gamma }\left(x\right)$, $\mathrm{\Gamma }\left(-x\right)$, $\ln \mathrm{\Gamma }\left(x\right)$, $\psi \left(x\right)$, $\psi \left(-x\right)$, ${\psi }^{\prime }\left(x\right)$, and ${\psi }^{\prime }\left(-x\right)$ for $x=0(.1)5$ to 8D or 8S; $\mathrm{\Gamma }\left(n+1\right)$ for $n=0(1)100(10)250(50)500(100)3000$ to 51S. … ►Abramov (1960) tabulates $\ln \mathrm{\Gamma }\left(x+\mathrm{i}y\right)$ for $x=1$ ($.01$) $2$, $y=0$ ($.01$) $4$ to 6D. Abramowitz and Stegun (1964, Chapter 6) tabulates $\ln \mathrm{\Gamma }\left(x+\mathrm{i}y\right)$ for $x=1$ ($.1$) $2$, $y=0$ ($.1$) $10$ to 12D. …Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of $\mathrm{\Gamma }\left(x+\mathrm{i}y\right)$, $\ln \mathrm{\Gamma }\left(x+\mathrm{i}y\right)$, and $\psi \left(x+\mathrm{i}y\right)$ for $x=0.5,1,5,10$, $y=0(.5)10$ to 8S.

14: 24.20 Tables

§24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. …

15: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Zhang and Jin (1996, Table 3.8) tabulates $\gamma (a,x)$ for $a=0.5,1,3,5,10,25,50,100$, $x=0(.1)1(1)3,5(5)30,50,100$ to 8D or 8S.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

16: 24.2 Definitions and Generating Functions

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§24.2(ii) Euler Numbers and Polynomials

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§24.2(iii) Periodic Bernoulli and Euler Functions

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Table 24.2.1: Bernoulli and Euler numbers.

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$n$	$B_n$	$E_n$
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Table 24.2.2: Bernoulli and Euler polynomials.

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$n$	$B_n\left(x\right)$	$E_n\left(x\right)$
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17: 25.15 Dirichlet $L$-functions

… ►For the principal character ${\chi }_1(modk)$, $L(s,{\chi }_1)$ is analytic everywhere except for a simple pole at $s=1$ with residue $\varphi \left(k\right)/k$, where $\varphi \left(k\right)$ is Euler’s totient function (§27.2). … ►

25.15.5 $$L(1-s,\chi )=\frac{k^{s-1}\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}+\chi (-1){\mathrm{e}}^{\pi \mathrm{i}s/2}\right)G(\chi )L(s,\overline{\chi }),$$

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

connection formula

Source:

Apostol (1976, (16), p. 263)

Referenced by:

§25.15(ii)

Permalink:

http://dlmf.nist.gov/25.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

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18: 20 Theta Functions

Chapter 20 Theta Functions

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19: 30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

►As ${\gamma }^2\to +\mathrm{\infty }$, with $q=2(n-m)+1$, … ►The asymptotic behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ and $a_{n,k}^m({\gamma }^2)$ as $n\to \mathrm{\infty }$ in descending powers of $2n+1$ is derived in Meixner (1944). …The asymptotic behavior of ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ and ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$ as $x\to \pm 1$ is given in Erdélyi et al. (1955, p. 151). The behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ for complex ${\gamma }^2$ and large $|{\lambda }_n^m\left({\gamma }^2\right)|$ is investigated in Hunter and Guerrieri (1982). …

20: 15.10 Hypergeometric Differential Equation

… ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: ►

15.10.17 $$w_3(z)=\frac{\mathrm{\Gamma }\left(1-c\right)\mathrm{\Gamma }\left(a+b-c+1\right)}{\mathrm{\Gamma }\left(a-c+1\right)\mathrm{\Gamma }\left(b-c+1\right)}{w}_1(z)+\frac{\mathrm{\Gamma }\left(c-1\right)\mathrm{\Gamma }\left(a+b-c+1\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{w}_2(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_2(z)$: Kummer’s solution $w_2$ and $w_3(z)$: Kummer’s solution $w_3$

Permalink:

http://dlmf.nist.gov/15.10.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.18 $$w_4(z)=\frac{\mathrm{\Gamma }\left(1-c\right)\mathrm{\Gamma }\left(c-a-b+1\right)}{\mathrm{\Gamma }\left(1-a\right)\mathrm{\Gamma }\left(1-b\right)}{w}_1(z)+\frac{\mathrm{\Gamma }\left(c-1\right)\mathrm{\Gamma }\left(c-a-b+1\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_2(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_2(z)$: Kummer’s solution $w_2$ and $w_4(z)$: Kummer’s solution $w_4$

Permalink:

http://dlmf.nist.gov/15.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.21 $$w_1(z)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(c-a-b\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_3(z)+\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a+b-c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{w}_4(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_3(z)$: Kummer’s solution $w_3$ and $w_4(z)$: Kummer’s solution $w_4$

Referenced by:

§15.8(i)

Permalink:

http://dlmf.nist.gov/15.10.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.25 $$w_1(z)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(b-a\right)}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)}{w}_5(z)+\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a-b\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_6(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_5(z)$: Kummer’s solution $w_5$ and $w_6(z)$: Kummer’s solution $w_6$

Referenced by:

§15.8(i)

Permalink:

http://dlmf.nist.gov/15.10.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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