Dirichlet series information

In mathematics, a Dirichlet series is any series of the form ∑ n = 1 ∞ a n n s , {\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}},} where s is...

In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of ∑ n = 1 ∞ a n e − λ n s , {\displaystyle...

expansion on an annulus centered at the singularity. A general Dirichlet series is a series of the form ∑ n = 1 ∞ a n e − λ n s . {\textstyle \sum _{n=1}^{\infty...

naturally in the study of Dirichlet series such as the Riemann zeta function. It describes the multiplication of two Dirichlet series in terms of their coefficients:...

functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence...

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

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

Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known. The Riemann zeta function...

1 if n = 1 and 0 otherwise, sometimes called multiplication unit for Dirichlet convolution or simply the unit function (completely multiplicative). Sometimes...

any sums over an arithmetic function f(n). Such sums are known as Dirichlet series. The regularized form f ~ ( s ) = ∑ n = 1 ∞ f ( n ) n − s {\displaystyle...

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

In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted Dir ⁡ ( α ) {\displaystyle \operatorname...

In analytic number theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic...

define the sum of the divergent series by the limit above. A series of this type is known as a generalized Dirichlet series; in applications to physics,...

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...

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

transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the...

cross over, in which case the series will diverge. An analogous concept is the abscissa of convergence of a Dirichlet series ∑ n = 1 ∞ a n n s . {\displaystyle...

The von Mangoldt function plays an important role in the theory of Dirichlet series, and in particular, the Riemann zeta function. For example, one has...

isomorphism from the reduced incidence algebra to the ring of formal Dirichlet series by sending δ n {\displaystyle \delta _{n}} to n − s , {\displaystyle...

In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions...

hyperbola method Dirichlet integral Dirichlet kernel (functional analysis, Fourier series) Dirichlet L-function Dirichlet principle Dirichlet problem (partial...

In mathematics, the Dirichlet function is the indicator function 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} of the set of rational numbers Q {\displaystyle...

A first step in this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy the relation ( 1 − 2 2 s ) ζ (...