Dirichlet%20L-functions

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11: 20 Theta Functions

Chapter 20 Theta Functions

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12: 25.21 Software

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§25.21(ix) Dirichlet $L$-series

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13: 27.9 Quadratic Characters

§27.9 Quadratic Characters

… ►The Legendre symbol $(n|p)$, as a function of $n$, is a Dirichlet character (mod $p$). … ►The Jacobi symbol $(n|P)$ is a Dirichlet character (mod $P$). …

14: 23.2 Definitions and Periodic Properties

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23.2.5 $$\zeta \left(z\right)=\frac{1}{z}+\sum _{w\in 𝕃\setminus \{0\}}\left(\frac{1}{z-w}+\frac{1}{w}+\frac{z}{w^2}\right),$$

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Defines:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function

Symbols:

$g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\zeta (z;g_2,g_3)$: Weierstrass zeta function, $\in$: element of, $\setminus$: set subtraction, $𝕃$: lattice and $z$: complex

Keywords:

modular functions, Weierstrass zeta function

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.2(ii), §23.2 and Ch.23

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23.2.6 $$\sigma \left(z\right)=z\prod _{w\in 𝕃\setminus \{0\}}\left(\left(1-\frac{z}{w}\right)\exp \left(\frac{z}{w}+\frac{z^2}{2w^2}\right)\right).$$

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Defines:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function

Symbols:

$g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\sigma (z;g_2,g_3)$: Weierstrass sigma function, $\in$: element of, $\exp z$: exponential function, $\setminus$: set subtraction, $𝕃$: lattice and $z$: complex

Keywords:

modular functions, Weierstrass sigma function

Referenced by:

§23.2(iii)

Permalink:

http://dlmf.nist.gov/23.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.2(ii), §23.2 and Ch.23

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15: 27.11 Asymptotic Formulas: Partial Sums

§27.11 Asymptotic Formulas: Partial Sums

… ►For example, Dirichlet (1849) proves that for all $x\ge 1$, …Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number ${\theta }_0$ such that the error term in (27.11.2) is $O\left(x^{\theta }\right)$ for all $\theta >{\theta }_0$. … ►where $\left(h,k\right)=1$, $k>0$. ►Letting $x\to \mathrm{\infty }$ in (27.11.9) or in (27.11.11) we see that there are infinitely many primes $p\equiv h(modk)$ if $h,k$ are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions. …

16: Bibliography G

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Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.

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External Links:

MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

Encodings:

BibTeX

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.

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External Links:

ISSN 0026-9255, MathReview (Peter Stevenhagen), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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17: 25.11 Hurwitz Zeta Function

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Figure 25.11.1: Hurwitz zeta function $\zeta (x,a)$, $a$ = 0. …8, 1, $-20\le x\le 10$. … Magnify

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25.11.36Removed because it is just (25.15.1) combined with (25.15.3).

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function and $k$: nonnegative integer

Referenced by:

§25.11(x), Erratum (V1.0.23) for Equation (25.11.36), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.11.E36

Removal (effective with 1.1.4):

This equation has been removed because it is just (25.15.1) combined with (25.15.3).

Clarification (effective with 1.0.23):

The Dirichlet $L$-function was inserted on the left-hand side. The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

See also:

Annotations for §25.11(x), §25.11 and Ch.25

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18: 24.16 Generalizations

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§24.16(ii) Character Analogs

►Let $\chi$ be a primitive Dirichlet character $modf$ (see §27.8). Then $f$ is called the conductor of $\chi$. … ►

24.16.11 $$B_{n,\chi }(x)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_{k,\chi }{x}^{n-k}.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer, $x$: real or complex, $\chi (a)$: Dirichlet character and $B_{n,\chi }$: Generalized Bernoulli numbers

Permalink:

http://dlmf.nist.gov/24.16.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(ii), §24.16 and Ch.24

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24.16.12 $$B_n\left(x\right)=B_{n,{\chi }_0}(x-1),$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $n$: integer, $x$: real or complex, $\chi (a)$: Dirichlet character and $B_{n,\chi }$: Generalized Bernoulli numbers

Referenced by:

§24.16(ii)

Permalink:

http://dlmf.nist.gov/24.16.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(ii), §24.16 and Ch.24

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19: Software Index

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	Open Source													With Book						Commercial
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20 Theta Functions
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25.21(ix) Dirichlet $L$-series	✓											a	✓									✓
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20: Bibliography M

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Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.

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External Links:

Document, MathReview (K.-B. Gundlach), ZentralBlatt

Referenced by:

§35.4(ii).

Encodings:

BibTeX

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

BibTeX

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:

FullText

Referenced by:

§8.24(i).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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