In mathematics, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as Z ( V , s...

zeta function is (usually) a function analogous to the original example, the Riemann zeta function ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum...

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...

the Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical...

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

of which is still conjectural. A prototypical example, the Riemann zeta function has a functional equation relating its value at the complex number s...

In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p2, p3, and so on....

sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding...

arithmetic point of view (including the Fermat varieties). Their local zeta-functions are computed in terms of Jacobi sums. Waring's problem is the most...

the roots of the local zeta-function of E. In this form it can be seen to be the analogue of the Riemann hypothesis for the function field associated...

Zeta (UK: /ˈziːtə/, US: /ˈzeɪtə/ ; uppercase Ζ, lowercase ζ; Ancient Greek: ζῆτα, Demotic Greek: ζήτα, classical [d͡zɛ̌ːta] or [zdɛ̌ːta] zē̂ta; Modern...

analysis to local zeta functions, and in particular for a proof of the first part of the Weil conjectures: the rationality of the zeta function of a variety...

understood and proven with the help of some general theory; see local zeta function and étale cohomology for example. The set of points E(Fq) is a finite...

In algebraic geometry, the motivic zeta function of a smooth algebraic variety X {\displaystyle X} is the formal power series: Z ( X , t ) = ∑ n = 0 ∞...

(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...

number theory. The conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over...

Dwork proved one of the four Weil conjectures (rationality of the local zeta function) in 1960. Grothendieck developed étale cohomology theory to prove...

Number of Primes Less Than a Given Magnitude", in which he defined his zeta function and proved its basic properties. The problem is named after the city...

{\displaystyle Y^{2}=X(X-1)(X-2)\ldots (X-k)} . Bounds for the zeroes of the local zeta-function immediately imply bounds for sums ∑ χ ( X ( X − 1 ) ( X − 2 ) … (...

polygamma function of order 2 k {\displaystyle 2k} . The Riemann–Siegel theta function is of interest in studying the Riemann zeta function, since it...

Elliott–Halberstam conjecture Functional equation (L-function) Chebotarev's density theorem Local zeta function Weil conjectures Modular form modular group Congruence...

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...

connected to the local zeta-function of a conic section. More generally, such sums for the Jacobi symbol relate to local zeta-functions of elliptic curves...

The solution is elementary (as we would now say, it computes a local zeta-function, for a curve that is a conic). One has (P − P*)2 = p or −p, for p...

that the local zeta-function of C has a factorization; this is the Artin L-function theory for the case of global fields that are function fields, for...

Segal and G. Smith showed that the local zeta function ζ G , p ( s ) = ∑ ν = 0 ∞ s p n ( G ) p − n s {\displaystyle \zeta _{G,p}(s)=\sum _{\nu =0}^{\infty...

group of ideles to lift the zeta function twisted by a Hecke character, i.e. a Hecke L-function, of a number field to a zeta integral and study its properties...

influential proposals by André Weil (1949) on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic...

application of p-adic numbers to local class field theory and diophantine geometry (Hasse principle), and to local zeta functions. Hasse was born in Kassel,...

divisibility result for the (reciprocals of) the zeroes and poles of the local zeta-function. Namely, the same power of q {\displaystyle q} divides each of these...