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11: 27.8 Dirichlet Characters

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27.8.4 $$\chi \left(n\right)=0,$$ $\left(n,k\right)>1$.

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Symbols:

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

Permalink:

http://dlmf.nist.gov/27.8.E4

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Annotations for §27.8 and Ch.27

… ►A divisor $d$ of $k$ is called an induced modulus for $\chi$ if …

12: 27.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►►►►

$d,k,m,n$	positive integers (unless otherwise indicated).
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$\left(m,n\right)$	greatest common divisor of $m,n$. If $\left(m,n\right)=1$, $m$ and $n$ are called relatively prime, or coprime.
$\left(d_1,\mathrm{\dots },d_n\right)$	greatest common divisor of $d_1,\mathrm{\dots },d_n$.
${\sum }_{d|n}$, ${\prod }_{d|n}$	sum, product taken over divisors of $n$.
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$p,p_1,p_2,\mathrm{\dots }$	prime numbers (or primes): integers ($>1$) with only two positive integer divisors, $1$ and the number itself.
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13: 25.15 Dirichlet $L$-functions

§25.15 Dirichlet $L$-functions

… ► … ►where ${\chi }_0$ is a primitive character (mod $d$) for some positive divisor $d$ of $k$ (§27.8). ►When $\chi$ is a primitive character (mod $k$) the $L$-functions satisfy the functional equation: … ►

§25.15(ii) Zeros

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14: 24.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►

$j,k,\mathrm{\ell },m,n$	integers, nonnegative unless stated otherwise.
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$(k,m)$	greatest common divisor of $k,m$.
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15: 27.7 Lambert Series as Generating Functions

§27.7 Lambert Series as Generating Functions

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27.7.3 $$\sum _{n=1}^{\mathrm{\infty }}\mu \left(n\right)\frac{x^n}{1-x^n}=x,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.1 I.B

Permalink:

http://dlmf.nist.gov/27.7.E3

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Annotations for §27.7 and Ch.27

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27.7.5 $$\sum _{n=1}^{\mathrm{\infty }}{n}^{\alpha }\frac{x^n}{1-x^n}=\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{\alpha }\left(n\right)x^n,$$

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 I.B

Permalink:

http://dlmf.nist.gov/27.7.E5

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Annotations for §27.7 and Ch.27

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27.7.6 $$\sum _{n=1}^{\mathrm{\infty }}\lambda \left(n\right)\frac{x^n}{1-x^n}=\sum _{n=1}^{\mathrm{\infty }}{x}^{n^2}.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/27.7.E6

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Annotations for §27.7 and Ch.27

16: 26.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►

$x$	real variable.
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$\left(h,k\right)$	greatest common divisor of positive integers $h$ and $k$.

►The main functions treated in this chapter are: ► ►►►►

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$	binomial coefficient.
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$p\left(n\right)$	number of partitions of $n$.
$p_k\left(n\right)$	number of partitions of $n$ into at most $k$ parts.
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17: 26.9 Integer Partitions: Restricted Number and Part Size

… ► $p_k\left(n\right)$ denotes the number of partitions of $n$ into at most $k$ parts. See Table 26.9.1. … ►

§26.9(ii) Generating Functions

… ►where the inner sum is taken over all positive divisors of $t$ that are less than or equal to $k$. …

18: 26.10 Integer Partitions: Other Restrictions

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26.10.18

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\left(m,n\right)$: greatest common divisor (gcd), $\mathrm{i}$: imaginary unit, $h$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer, $A_k(n)$: coefficient and $f(h,k)$: function

Permalink:

http://dlmf.nist.gov/26.10.E18

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Annotations for §26.10(vi), §26.10 and Ch.26

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19: 23.18 Modular Transformations

§23.18 Modular Transformations

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Elliptic Modular Function

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Dedekind’s Eta Function

… ►where the square root has its principal value and … ►

23.18.7 $$s(d,c)=\sum _{r=1}^{c-1}\frac{r}{c}\left(\frac{dr}{c}-\lfloor \frac{dr}{c}\rfloor -\frac{1}{2}\right),$$ $c>0$.

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Symbols:

$\lfloor x\rfloor$: floor of $x$ , $\left(m,n\right)$: greatest common divisor (gcd), $c$: integer and $d$: integer

Referenced by:

§23.18, Erratum (V1.0.11) for Equation (23.18.7)

Permalink:

http://dlmf.nist.gov/23.18.E7

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Errata (effective with 1.0.11):

Originally the sum ${\sum }_{r=1}^{c-1}$ was written with an additional constraint on the summation, that $\left(r,c\right)=1$. This additional condition was incorrect and has been removed.

Reported 2015-10-05 by Howard Cohl and Tanay Wakhare

See also:

Annotations for §23.18, §23.18 and Ch.23

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20: Bibliography K

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On the evaluation of the Gauss hypergeometric function. C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.

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External Links:

ISSN 0861-1459, MathReview, ZentralBlatt

Referenced by:

§15.15, §15.19(i).

Encodings:

BibTeX

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Generalized functions. Mathematics in Science and Engineering, Vol. 171, Academic Press, Inc., Orlando, FL.

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Notes:

Theory and technique

External Links:

ISBN 0-12-396560-8, MathReview (Qing Yi Chen), ZentralBlatt

Referenced by:

§1.16(viii).

Encodings:

BibTeX

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Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.13(iii), §8.22(ii).

Encodings:

BibTeX

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Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Comput. Phys. Comm. 4, pp. 221–226.

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Notes:

Single-precision Fortran, accuracy 12-14S

External Links:

Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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An improvement of the remainder term in the divisor problem. Mat. Zametki 6, pp. 545–554 (Russian).

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Notes:

In Russian. English translation: Math. Notes 6(1969), no. 5, pp. 784–791.

External Links:

Document, MathReview (A. I. Vinogradov), ZentralBlatt

Referenced by:

§27.11, Erratum (V1.1.8) for Section 27.11.

Encodings:

BibTeX

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