# Dirichlet theorem

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##### 1: 27.11 Asymptotic Formulas: Partial Sums
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. …
##### 2: 25.15 Dirichlet $L$-functions
This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions (§27.11). Related results are:
25.15.10 $L\left(0,\chi\right)=\begin{cases}\displaystyle-\frac{1}{k}\sum_{r=1}^{k-1}r% \chi(r),&\chi\neq\chi_{1},\\ 0,&\chi=\chi_{1}.\end{cases}$
##### 3: 27.8 Dirichlet Characters
For any character $\chi\pmod{k}$, $\chi\left(n\right)\neq 0$ if and only if $\left(n,k\right)=1$, in which case the Euler–Fermat theorem (27.2.8) implies $\left(\chi\left(n\right)\right)^{\phi\left(k\right)}=1$. …
##### 4: 27.4 Euler Products and Dirichlet Series
###### §27.4 Euler Products and Dirichlet Series
The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. … called Dirichlet series with coefficients $f(n)$. The function $F(s)$ is a generating function, or more precisely, a Dirichlet generating function, for the coefficients. …
##### 5: Bibliography
• T. M. Apostol (1952) Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.
• T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.
• T. M. Apostol (1985b) Note on the trivial zeros of Dirichlet $L$-functions. Proc. Amer. Math. Soc. 94 (1), pp. 29–30.
• T. M. Apostol (1990) Modular Functions and Dirichlet Series in Number Theory. 2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.
• T. M. Apostol (2000) A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.
• ##### 6: Bibliography G
• G. Gasper (1975) Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.
• K. Girstmair (1990a) A theorem on the numerators of the Bernoulli numbers. Amer. Math. Monthly 97 (2), pp. 136–138.
• K. Girstmair (1990b) Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.
• R. A. Gustafson (1987) Multilateral summation theorems for ordinary and basic hypergeometric series in ${\rm U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.
• ##### 7: 25.2 Definition and Expansions
For further expansions of functions similar to (25.2.1) (Dirichlet series) see §27.4. …
25.2.8 $\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N% }^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{s+1}}\mathrm{d}x,$ $\Re s>0$, $N=1,2,3,\dots$.
##### 8: Bibliography D
• S. C. Dhar (1940) Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.
• H. Ding, K. I. Gross, and D. St. P. Richards (1996) Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.
• 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).
• J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.
##### 10: Errata
• Equation (25.15.10)
25.15.10 $L\left(0,\chi\right)=\begin{cases}\displaystyle-\frac{1}{k}\sum_{r=1}^{k-1}r% \chi(r),&\chi\neq\chi_{1},\\ 0,&\chi=\chi_{1}.\end{cases}$

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

• Paragraph Prime Number Theorem (in §27.12)

The largest known prime, which is a Mersenne prime, was updated from $2^{43,112,609}-1$ (2009) to $2^{82,589,933}-1$ (2018).

• Equation (33.14.15)
33.14.15 $\int_{0}^{\infty}\phi_{m,\ell}(r)\phi_{n,\ell}(r)\mathrm{d}r=\delta_{m,n}$

The definite integral, originally written as $\int_{0}^{\infty}\phi_{n,\ell}^{2}(r)\mathrm{d}r=1$, was clarified and rewritten as an orthogonality relation. This follows from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2).

• Equation (25.11.36)

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).