# Dirichlet L-functions

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## 11—20 of 34 matching pages

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 $P$). …
##### 12: 27.11 Asymptotic Formulas: Partial Sums
###### §27.11 Asymptotic Formulas: Partial Sums
For example, Dirichlet (1849) proves that for all $x\geq 1$, …Dirichlet’s divisor problem (unsolved in 2009) 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 $x\to\infty$ in (27.11.9) or in (27.11.11) we see that there are infinitely many primes $p\equiv h\pmod{k}$ if $h,k$ are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions. …
##### 13: 24.16 Generalizations
###### §24.16(ii) Character Analogs
Let $\chi$ be a primitive Dirichlet character $\mod f$ (see §27.8). Then $f$ is called the conductor of $\chi$. …
24.16.11 $B_{n,\chi}(x)=\sum_{k=0}^{n}{n\choose k}B_{k,\chi}x^{n-k}.$
##### 14: 23.10 Addition Theorems and Other Identities
23.10.5 $\begin{vmatrix}1&\wp\left(u\right)&\wp'\left(u\right)\\ 1&\wp\left(v\right)&\wp'\left(v\right)\\ 1&\wp\left(w\right)&\wp'\left(w\right)\end{vmatrix}=0,$
23.10.6 $\left(\zeta\left(u\right)+\zeta\left(v\right)+\zeta\left(w\right)\right)^{2}+% \zeta'\left(u\right)+\zeta'\left(v\right)+\zeta'\left(w\right)=0.$
##### 15: 30.15 Signal Analysis
The sequence $\phi_{n}$, $n=0,1,2,\dots$ forms an orthonormal basis in the space of $\sigma$-bandlimited functions, and, after normalization, an orthonormal basis in $L^{2}(-\tau,\tau)$. …
##### 16: 10.43 Integrals
10.43.2 $\int z^{\nu}\mathscr{Z}_{\nu}\left(z\right)\mathrm{d}z=\pi^{\frac{1}{2}}2^{\nu% -1}\Gamma\left(\nu+\tfrac{1}{2}\right)z\*\left(\mathscr{Z}_{\nu}\left(z\right)% \mathbf{L}_{\nu-1}\left(z\right)-\mathscr{Z}_{\nu-1}\left(z\right)\mathbf{L}_{% \nu}\left(z\right)\right),$ $\nu\neq-\tfrac{1}{2}$.
##### 18: Bibliography D
• C. de la Vallée Poussin (1896b) 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).
• P. G. L. Dirichlet (1837) 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).
• P. G. L. Dirichlet (1849) Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).
• ##### 19: 30.13 Wave Equation in Prolate Spheroidal Coordinates
###### §30.13(v) The Interior Dirichlet Problem for Prolate Ellipsoids
For the Dirichlet boundary-value problem of the region $\xi_{1}\leq\xi\leq\xi_{2}$ between two ellipsoids, the eigenvalues are determined from …
##### 20: 27.3 Multiplicative Properties
Examples are $\left\lfloor 1/n\right\rfloor$ and $\lambda\left(n\right)$, and the Dirichlet characters, defined in §27.8. …