mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes...

{1}{n^{s}}}.} Zeta functions include: Airy zeta function, related to the zeros of the Airy function Arakawa–Kaneko zeta function Arithmetic zeta function Artin–Mazur...

the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is a meromorphic function on the complex plane...

Unsolved problem in mathematics Do all non-trivial zeroes of the Riemann zeta function have a real part of one half? More unsolved problems in mathematics...

ln(x) or loge(x). In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive...

p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose...

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined...

the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained...

In number theory, a multiplicative function is an arithmetic function f {\displaystyle f} of a positive integer n {\displaystyle n} with the property that...

Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that...

In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent...

theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number...

partition function is the Riemann zeta function. This idea underlies Alain Connes's attempted proof of the Riemann hypothesis. The Möbius function is multiplicative...

(z)=\zeta _{H}'(0,z)-\zeta '(0),} where ζ H {\displaystyle \zeta _{H}} is the Hurwitz zeta function, ζ {\displaystyle \zeta } is the Riemann zeta function...

Dirichlet's theorem on arithmetic progressions. The Dirichlet series for φ(n) may be written in terms of the Riemann zeta function as: ∑ n = 1 ∞ φ ( n )...

numbers to what is known now as the Riemann zeta function: he showed that the value ζ ( 1 ) {\displaystyle \zeta (1)} reduces to a ratio of two infinite products...

The Liouville lambda function, denoted by λ(n) and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if n is the product...

{a_{n}}{n^{s}}}=\zeta (s)\sum _{n=1}^{\infty }{\frac {b_{n}}{n^{s}}}} where ζ(s) is the Riemann zeta function. Given an arithmetic function, one can generate...

arithmetic schemes. I". Documenta Mathematica: 261–284. ISBN 978-3-936609-21-9. Fesenko, I. (2008). "Adelic study of the zeta function of arithmetic schemes...

properties of the Riemann zeta function introduced by Riemann in 1859. Proofs of the prime number theorem not using the zeta function or complex analysis were...

arithmetic function is some simpler or better-understood function which takes the same values "on average". Let f {\displaystyle f} be an arithmetic function...

These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in (Dirichlet 1837) to prove the theorem on primes in arithmetic progressions...

J. Tate, Algebraic cycles and poles of zeta functions in the volume (O. F. G. Schilling, editor), Arithmetical Algebraic Geometry, pages 93–110 (1965)...

Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results...

definition of local zeta-function available. To get an L-function for A itself, one takes a suitable Euler product of such local functions; to understand the...

the verification of the hypothesis yet"). The Riesz function is related to the Riemann zeta function via its Mellin transform. If we take M ( R i e s z...

Synchrotron function Riemann zeta function: A special case of Dirichlet series. Riemann Xi function Dirichlet eta function: An allied function. Dirichlet...

zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar...

higher-dimensional generalization of the Goss zeta function. Goss, David (1996), Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer...

Dirichlet generating function of μ {\displaystyle \mu } is 1 / ζ ( s ) {\displaystyle 1/\zeta (s)} and the Dirichlet generating function of n k {\displaystyle...