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11: 27.10 Periodic Number-Theoretic Functions

… ►Examples are the Dirichlet characters (mod $k$) and the greatest common divisor $\left(n,k\right)$ regarded as a function of $n$. … ►Another generalization of Ramanujan’s sum is the Gauss sum $G(n,\chi )$ associated with a Dirichlet character $\chi (modk)$. It is defined by the relation … ►For any Dirichlet character $\chi (modk)$, $G(n,\chi )$ is separable for $n$ if $\left(n,k\right)=1$, and is separable for every $n$ if and only if $G(n,\chi )=0$ whenever $\left(n,k\right)>1$. … ►The finite Fourier expansion of a primitive Dirichlet character $\chi (modk)$ has the form …

12: 19.32 Conformal Map onto a Rectangle

… ►then $z(p)$ is a Schwartz–Christoffel mapping of the open upper-half $p$-plane onto the interior of the rectangle in the $z$-plane with vertices …

13: 27.5 Inversion Formulas

… ►If a Dirichlet series $F(s)$ generates $f(n)$, and $G(s)$ generates $g(n)$, then the product $F(s)G(s)$ generates ►

27.5.1 $$h(n)=\sum _{d|n}f(d)g\left(\frac{n}{d}\right),$$

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Symbols:

$d$: positive integer, $n$: positive integer, $f$: function, $g$: function and $h$: function

Permalink:

http://dlmf.nist.gov/27.5.E1

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Annotations for §27.5 and Ch.27

►called the Dirichlet product (or convolution) of $f$ and $g$. The set of all number-theoretic functions $f$ with $f(1)\ne 0$ forms an abelian group under Dirichlet multiplication, with the function $\lfloor 1/n\rfloor$ in (27.2.5) as identity element; see Apostol (1976, p. 129). … ►

27.5.6 $$G(x)=\sum _{n\le x}F\left(\frac{x}{n}\right)⟺F(x)=\sum _{n\le x}\mu \left(n\right)G\left(\frac{x}{n}\right),$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer, $x$: real number, $F(s)$: Dirichlet series and $G(s)$: Dirichlet series

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E6

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Annotations for §27.5 and Ch.27

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14: Bibliography D

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Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems. Constr. Approx. 40 (1), pp. 61–104.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (Francisco Marcellán), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

Encodings:

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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire $Mx+N$ . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

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Atomic motions in a rigid sphere gas as a problem in neutron transport. Nucl. Sci. Eng. 24 (2), pp. 142–152.

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Referenced by:

§18.39(iii).

Encodings:

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Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1837, pp. 45–81 (German).

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Notes:

Reprinted in G. Lejeune Dirichlet’s Werke, Band I, pp. 313–342, Reimer, Berlin, 1889 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.

External Links:

ISBN 0-8284-0225-6, Link, MathReview

Referenced by:

§25.15(i).

Encodings:

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Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).

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Notes:

Reprinted in G. Lejeune Dirichlet’s Werke, Band II, pp. 49–66, Reimer, Berlin, 1897 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.

External Links:

ISBN 0-8284-0225-6, Link, MathReview

Referenced by:

§27.11.

Encodings:

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15: Bibliography G

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Quasirandom distributed bases for bound problems. J. Chem. Phys. 114 (9), pp. 3929–3939.

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Referenced by:

§18.39(iii).

Encodings:

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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:

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Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

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Notes:

Problem proposed by P. Smith.

External Links:

Document

Referenced by:

(9.10.14), (9.10.15), (9.10.16), §9.10(v).

Encodings:

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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).

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Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:

FullText, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

Encodings:

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16: Bibliography J

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The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis. J. Reine Angew. Math. 583, pp. 29–86.

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External Links:

ISSN 0075-4102, Document, MathReview (Mehmet Can), ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

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The Painlevé connection problem: An asymptotic approach. I. Stud. Appl. Math. 86 (4), pp. 315–376.

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External Links:

ISSN 0022-2526, MathReview (I. P. Martynov), ZentralBlatt

Referenced by:

§32.11(i), §32.12(ii).

Encodings:

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17: 25.1 Special Notation

… ►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 $L(s,\chi )$.

18: 14.28 Sums

… ►where ${ℰ}_1$ and ${ℰ}_2$ are ellipses with foci at $\pm 1$, ${ℰ}_2$ being properly interior to ${ℰ}_1$. …

19: 29.19 Physical Applications

… ►Simply-periodic Lamé functions ($\nu$ noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. … ►

§29.19(ii) Lamé Polynomials

… ►Shail (1978) treats applications to solutions of elliptic crack and punch problems. …

20: 25.19 Tables

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Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

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Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta \left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta (s,a)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.