mathematics, a Dirichlet L {\displaystyle L} -series is a function of the form L ( s , χ ) = ∑ n = 1 ∞ χ ( n ) n s . {\displaystyle L(s,\chi )=\sum _{n=1}^{\infty...

It is a particular Dirichlet L-function, the L-function for the alternating character of period four. The Dirichlet beta function is defined as β ( s...

is a Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation. The Riemann zeta function is an...

Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. Specifically, the Riemann zeta function ζ(s) is the Dirichlet series of the...

in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number...

are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis...

(a)te^{at}}{e^{ft}-1}}} for χ a Dirichlet character with conductor f. The Kubota–Leopoldt p-adic L-function Lp(s, χ) interpolates the Dirichlet L-function with the Euler...

class of modified zeta functions and Dirichlet L-functions that possess exactly the same non-trivial zeros as the Riemann zeta function, but whose Euler products...

was proved by Dirichlet (1837) with Dirichlet L-series. The proof is modeled on Euler's earlier work relating the Riemann zeta function to the distribution...

mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced...

all over the prime numbers. Arithmetic function Dirichlet L-function Dirichlet series Multiplicative function Apostol, Tom (1976). Introduction to Analytic...

Dirichlet beta function Dirichlet L-function Hurwitz zeta function Legendre chi function Lerch transcendent Polylogarithm and related functions: Incomplete...

a complex-valued arithmetic function χ : Z → C {\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} } is a Dirichlet character of modulus m {\displaystyle...

Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known. The Riemann zeta function ζ(s) is a function of a complex...

on the left-hand side is also L ( 1 ) {\displaystyle L(1)} where L ( s ) {\displaystyle L(s)} is the Dirichlet L-function for the field of Gaussian rational...

begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions...

Johann Peter Gustav Lejeune Dirichlet (/ˌdɪərɪˈkleɪ/; German: [ləˈʒœn diʁiˈkleː]; 13 February 1805 – 5 May 1859) was a German mathematician. In number...

generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every...

missing. Explicit formula (L-function) Riemann–Siegel formula (particular approximate functional equation) "§25.15 Dirichlet -functions on NIST". Weisstein,...

Here y is a real parameter. The Riemann zeta function can be replaced by a Dirichlet L-function of a Dirichlet character χ. The sum over prime powers then...

the Hurwitz zeta function may be expressed as a linear combination of Dirichlet L-functions and vice versa: The Hurwitz zeta function coincides with Riemann's...

a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on...

tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function. The Clausen function of order 2 – often referred...

In mathematics, Dirichlet convolution (or divisor convolution) is a binary operation defined for arithmetic functions; it is important in number theory...

In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a complex-valued, periodic function f {\displaystyle f} to be equal to the sum...

the Gamma function. Such sums are ubiquitous in number theory. They occur, for example, in the functional equations of Dirichlet L-functions, where for...

generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting...

analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as D...

Sequences. 13: Article 10.7.4. Apostol, T. M. (1997), "Modular Functions and Dirichlet Series in Number Theory", New York: Springer-Verlag 2nd Ed. Ashworth...

the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over...