Dirichlet L-functions

SearchAdvancedHelp

(0.004 seconds)

21—29 of 29 matching pages

21: Bibliography J

… ►

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.

ⓘ

External Links:

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

Referenced by:

§32.11(i).

Encodings:

BibTeX

…

22: Bibliography G

… ►

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

ⓘ

External Links:

MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§24.16(iii).

Encodings:

BibTeX

…

23: 18.10 Integral Representations

… ►

§18.10(i) Dirichlet–Mehler-Type Integral Representations

…

24: 25.11 Hurwitz Zeta Function

… ►

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

ⓘ

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

…

25: Software Index

… ► ►►►

	Open Source													With Book						Commercial
…
25.21(ix) Dirichlet $L$-series	✓											a	✓									✓
…

…

26: 19.16 Definitions

… ►For generalizations and further information, especially representation of the $R$-function as a Dirichlet average, see Carlson (1977b). …

27: Bibliography M

… ►

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

ⓘ

External Links:

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

Referenced by:

§35.4(ii).

Encodings:

BibTeX

…

28: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

§1.18(ii) $L^2$ spaces on intervals in $ℝ$

… ►For $𝒟(T)$ we can take $C^2(X)$, with appropriate boundary conditions, and with compact support if $X$ is bounded, which space is dense in $L^2\left(X\right)$, and for $X$ unbounded require that possible non-$L^2$ eigenfunctions of (1.18.28), with real eigenvalues, are non-zero but bounded on open intervals, including $\pm \mathrm{\infty }$. … ►The implicit boundary conditions taken here are that the ${\varphi }_n(x)$ and ${\varphi }_n^{\prime }(x)$ vanish as $x\to \pm \mathrm{\infty }$, which in this case is equivalent to requiring ${\varphi }_n(x)\in L^2\left(X\right)$, see Pauling and Wilson (1985, pp. 67–82) for a discussion of this latter point. … ►Eigenfunctions corresponding to the continuous spectrum are non-$L^2$ functions. … ►The well must be deep and broad enough to allow existence of such $L^2$ discrete states. …

29: Errata

… ►

Immediately below (27.11.2), the bound ${\theta }_0$ for Dirichlet’s divisor problem (currently still unsolved) has been changed from $\frac{12}{37}$ Kolesnik (1969) to $\frac{131}{416}$ Huxley (2003).

… ►

25.15.10

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Reported by Gergő Nemes on 2021-08-23

… ►

We have emphasized the link with the Dirichlet $L$-function, and used the fact that $\chi (k)=0$. A sentence just below (25.11.36) was added indicating that one should make a comparison with (25.15.1) and (25.15.3).

…