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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
Local_zeta_function
Index of lists with the same name
the Riemann zeta function Local zeta function of a characteristic-p variety Matsumoto zeta function Minakshisundaram–Pleijel zeta function of a Laplacian
List_of_zeta_functions
Analytic function in mathematics
Riemann zeta function, or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ {\displaystyle \zeta } (zeta), is a function of a complex
Riemann_zeta_function
Mathematical function associated to algebraic varieties
global L-function defined as an Euler product of local zeta functions. Hasse–Weil L-functions form one of the two major classes of global L-functions, alongside
Hasse–Weil_zeta_function
Meromorphic function on the complex plane
L-functions share fundamental properties and characteristics with the Riemann zeta function, which serves as the prototypical example of an L-function;
L-function
Mathematical concept
sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Summability method in physics
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent
Zeta_function_regularization
Type of generating function in mathematics
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.
Igusa_zeta_function
of which is still conjectural. A prototypical example, the Riemann zeta function has a functional equation relating its value at the complex number s
Functional equation (L-function)
Functional_equation_(L-function)
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
Artin–Mazur_zeta_function
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
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Estimates the number of points on an elliptic curve over a finite field
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
Hasse's theorem on elliptic curves
Hasse's_theorem_on_elliptic_curves
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 ∞
Motivic_zeta_function
Conjecture on zeros of the zeta function
Unsolved problem in mathematics Do all non-trivial zeros of the Riemann zeta function have a real part equal to one half? More unsolved problems in mathematics
Riemann_hypothesis
Sixth letter in the Greek alphabet
Zeta (UK: /ˈziːtə/, US: /ˈzeɪtə/ ; uppercase Ζ, lowercase ζ; (Ancient Greek and Katharevousa: ζῆτα, Demotic Greek: ζήτα, classical [d͡zɛ̌ːta] or [zdɛ̌ːta]
Zeta
Extension of the factorial function
(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
Gamma_function
Number-theoretic concept
In general the values of Jacobi sums occur in relation with the local zeta-functions of diagonal forms. The result on the Legendre symbol amounts to the
Jacobi_sum
American mathematician
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
Bernard_Dwork
Algebraic curve in mathematics
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
Elliptic_curve
Mathematical construct
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
Character_sum
English mathematician (1907–1969)
{\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 ) … (
Harold_Davenport
Hyperspace in algebraic geometry
201–223 Bombieri, Enrico; Swinnerton-Dyer, H. P. F. (1967), "On the local zeta function of a cubic threefold", Ann. Scuola Norm. Sup. Pisa (3), 21: 1–29
Cubic_threefold
On generating functions from counting points on algebraic varieties over finite fields
number theory. The conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over
Weil_conjectures
Mathematic theory
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
Tate's_thesis
Mathematical function
polygamma function of order 2 k {\displaystyle 2k} . The Riemann–Siegel theta function is of interest in studying the Riemann zeta function, since it
Riemann–Siegel_theta_function
Branch of algebraic geometry
Dwork proved one of the four Weil conjectures (rationality of the local zeta function) in 1960. Grothendieck developed étale cohomology theory to prove
Arithmetic_geometry
German mathematician (1898–1979)
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,
Helmut_Hasse
Type of Dirichlet series associated to number field extensions
theory and are generalizations of better-known functions like Dedekind zeta functions or Dirichlet L-functions. Some of their expected properties turned out
Artin_L-function
Particular kind of exponential sum
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
Kloosterman_sum
Sum of inverse squares of natural numbers
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
Basel_problem
Type of homogenous polynomial
deal has been worked out about their theory: algebraic geometry, local zeta-functions via Jacobi sums, Hardy-Littlewood circle method. Over a field of
Diagonal_form
Great circle with a characteristic length
Hirzebruch–Riemann–Roch theorem Local zeta function Measurable Riemann mapping theorem Riemann (crater) Riemann Xi function Riemann curvature tensor Riemann
Metric_circle
Mathematical term; concerning axioms used to derive theorems
consistency, in many cases independence is not a necessary requirement for a functioning axiomatic system — though it is usually sought after to minimize the
Axiomatic_system
Certain polynomial equations in enough variables over a finite field have solutions
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
Chevalley–Warning_theorem
Proposition in mathematics that is unproven
influential proposals by André Weil (1949) on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic
Conjecture
Elliott–Halberstam conjecture Functional equation (L-function) Chebotarev's density theorem Local zeta function Weil conjectures Modular form modular group Congruence
List_of_number_theory_topics
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
Gaussian_period
Type of mathematical functions
in this field. Patching the local data of meromorphic functions, i.e. the problem of creating a global meromorphic function from zeros and poles, is called
Function of several complex variables
Function_of_several_complex_variables
Concept in number theory (mathematics)
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
Arithmetic of abelian varieties
Arithmetic_of_abelian_varieties
Mathematical function
-\sum _{k=1}^{\infty }(-1)^{k}\,\zeta (k+1)\,z^{k},} which converges for |z| < 1. Here, ζ(n) is the Riemann zeta function. This series is easily derived
Digamma_function
mathematics, motivic L-functions are a generalization of Hasse–Weil L-functions to general motives over global fields. The local L-factor at a finite place
Motivic_L-function
Unsolved problem in mathematics
Hypothesis for local zeta functions) by Deligne (1974). Ramanujan's original hypothesis was inspired by his research on a particular L-function, nowadays called
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Special function in the physical sciences
until his retirement in 1881. Mathematics portal Physics portal Airy zeta function Aspnes, David E. (1966). "Electric-Field Effects on Optical Absorption
Airy_function
Mexican criminal syndicate
Los Zetas (pronounced [los ˈsetas], Spanish for "The Zs") is a fractured Mexican criminal syndicate and designated terrorist organization, known as one
Los_Zetas
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
Subgroup_growth
Family of hypersurfaces in algebraic geometry
Bernard Dwork. Originally considered by Dwork in the context of local zeta-functions, such families have been shown to have relationships with mirror
Dwork_family
Serbian mathematician and university teacher
gained an international reputation and gave lectures on the Riemann zeta function at universities around the world. Aleksandar Ivić was born in Belgrade
Aleksandar_Ivić
French mathematician (1928–2014)
geometry. They describe properties of analytic invariants, called local zeta functions, of the number of points on an algebraic curve or variety of higher
Alexander_Grothendieck
Type of surface in algebraic geometry
theory Bombieri, Enrico; Swinnerton-Dyer, H. P. F. (1967), "On the local zeta function of a cubic threefold", Ann. Scuola Norm. Sup. Pisa (3), 21: 1–29
Fano_surface
Sheaf cohomology on the étale site
étale cohomology. This is how the theory could be applied to the local zeta-function of an algebraic curve. Theorem. Let X be a curve of genus g defined
Étale_cohomology
version of the functional equation for the Riemann zeta function. This involves giving the zeta function of a number field an integral representation in
Schwartz–Bruhat_function
American animated television series
The Zeta Project is an American animated science fiction television series produced by Warner Bros. Television Animation, which originally aired on Kids'
The_Zeta_Project
Russian mathematician
objects. He pioneered the study of zeta functions in higher dimensions by developing his theory of higher adelic zeta integrals. These integrals are defined
Ivan_Fesenko
Type of character in number theory
to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have
Hecke_character
Function in fluid mathematics
and ζ = z L {\displaystyle \zeta ={\dfrac {z}{L}}} From there, a function φ M ( ζ ) {\displaystyle \varphi _{M}(\zeta )} can be determined to empirically
Monin–Obukhov similarity theory
Monin–Obukhov_similarity_theory
Numbers expressible as integrals of algebraic functions
NT]. Belkale, Prakash; Brosnan, Patrick (2003). "Periods and Igusa local zeta functions". International Mathematics Research Notices. 2003 (49): 2655. doi:10
Period_(number_theory)
Function equal to the product of its values on coprime factors
_{P}(1-|P|^{-s})^{-1}.} Unlike the classical zeta function, ζ A ( s ) {\displaystyle \zeta _{A}(s)} is a simple rational function: ζ A ( s ) = ∑ f | f | − s = ∑ n
Multiplicative_function
Number of integers coprime to and less than n
Riemann zeta function as: ∑ n = 1 ∞ φ ( n ) n s = ζ ( s − 1 ) ζ ( s ) {\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta
Euler's_totient_function
Topics referred to by the same term
Hasse–Weil zeta function. Gutzwiller trace formula: See Quantum chaos Kuznetsov trace formula, an extension of the Petersson trace formula. Local trace formula
Trace_formula
Mathematical concept
ISBN 978-3-540-17848-4, MR 0892097 Godement, Roger; Jacquet, Hervé (1972), Zeta Functions of Simple Algebras, Lecture Notes in Mathematics, vol. 260, Berlin,
Automorphic_L-function
Physical system that responds to a restoring force proportional to displacement
_{0}\zeta \right)^{2}+{\frac {1}{\omega ^{2}}}(\omega _{0}^{2}-\omega ^{2})^{2}}}} is the absolute value of the impedance or linear response function, and
Harmonic_oscillator
Mathematical seminars held in Paris since 1948
de fonctions algébriques de caractéristique p, I, d'après Weil (local zeta-function) Roger Godement, Groupe complexe unimodulaire, I : Les représentations
Séminaire_Nicolas_Bourbaki
Theorem in number theory
role in the Langlands program, by identifying a part of the Hasse–Weil zeta function of a modular curve or a more general modular variety, with the product
Eichler–Shimura congruence relation
Eichler–Shimura_congruence_relation
Mathematical function
exponential function is a p-adic analogue of the usual exponential function on the complex numbers. As in the complex case, it has an inverse function, named
P-adic_exponential_function
Set of functions used to represent the electronic wave function
functions def2-TZVPPD – Valence triple-zeta with two sets of polarization functions and a set of diffuse functions def2-QZVP – Valence quadruple-zeta
Basis_set_(chemistry)
Open subset of the real–number line
a geometric zeta function ζ L {\displaystyle \zeta _{\mathcal {L}}} : the Dirichlet series ζ L ( s ) = ∑ j ∈ J ℓ j s {\displaystyle \zeta _{\mathcal {L}}(s)=\sum
Fractal_string
Mathematics of varieties with integer coordinates
geometry along with class field theory, complex multiplication, local zeta-functions and L-functions. Paul Vojta wrote: While others at the time shared this viewpoint
Diophantine_geometry
Indian-American mathematician
(link) Belkale, Prakash; Brosnan, Patrick (2003). "Periods and Igusa Local Zeta functions". Int. Math. Res. Not. 2003 (49): 2655–2670. doi:10.1155/S107379280313142X
Prakash_Belkale
sorority, not the former local honor society for library science at Syracuse, now a part of Beta Phi Mu. Merged with Delta Zeta. Merged with Beta Sigma
List of social sororities and women's fraternities
List_of_social_sororities_and_women's_fraternities
knowing N modulo p determines N for p ≥ 5. This connection with local zeta-functions has been investigated in depth. For a plane curve defined by a cubic
Hasse–Witt_matrix
Medieval principality in south-east Europe
Zeta (Serbian Cyrillic: Зета; Albanian: Zeta; Latin: Zenta or Genta) was one of the medieval polities that existed between 1371 and 1421, whose territory
Zeta_under_the_Balšići
Japanese mathematician (1924–2013)
his contributions to algebraic geometry and number theory. The Igusa zeta-function, the Igusa quartic, Igusa subgroups, Igusa curves, and Igusa varieties
Jun-Ichi_Igusa
Association of one output to each input
complex function is illustrated by the multiplicative inverse of the Riemann zeta function: the determination of the domain of definition of the function z
Function_(mathematics)
Type of function in mathematics
the negative integers The Riemann zeta function except for a simple pole at 1 {\displaystyle 1} Algebraic functions are analytic away from any poles and
Analytic_function
Topology in the study of subharmonic functions
{\displaystyle \zeta } if there exists a subharmonic function v {\displaystyle v} defined on a neighbourhood of ζ {\displaystyle \zeta } such that v (
Fine topology (potential theory)
Fine_topology_(potential_theory)
Structure in algebraic geometry
points over any finite field, and in multiplicative notation for local zeta-functions. The general idea is that one motive has the same structure in any
Motive_(algebraic_geometry)
Programming language feature
documentation". "Anonymous Functions - MATLAB & Simulink - MathWorks United Kingdom". Partial Function Evaluation in MATLAB Closures in ZetaLisp Archived 2012-03-19
First-class_function
Generalized function whose value is zero everywhere except at zero
{1}{2\pi i}}\oint _{\partial D}{\frac {f(\zeta )\,d\zeta }{\zeta -z}},\quad z\in D} for all holomorphic functions f in D that are continuous on the closure
Dirac_delta_function
Russian mathematician (1937–2008)
Riemann zeta-function on the critical line". Proc. Steklov Inst. Math. (167): 167–178. Selberg, A. (1942). "On the zeros of Riemann's zeta-function". SHR
Anatoly_Karatsuba
Mathematical Theory
who exhibited the zeta function as the Mellin transform of Jacobi's theta function. Riemann used asymptotics of the theta function to obtain the analytic
Rankin–Selberg_method
Euler system p-adic L-function Arithmetic geometry Complex multiplication Abelian variety of CM-type Chowla–Selberg formula Hasse–Weil zeta function
List of algebraic number theory topics
List_of_algebraic_number_theory_topics
Infinite matrices with Pascal's triangle as elements
related to counting points on elliptic curves over finite fields and local zeta functions . A Pascal matrix can actually be constructed by taking the matrix
Pascal_matrix
Norwegian mathematician (1917–2007)
including a proof that a positive proportion of the zeros of the Riemann zeta function lie on the line ℜ ( s ) = 1 2 {\displaystyle \Re (s)={\tfrac {1}{2}}}
Atle_Selberg
German mathematician (born 1958)
Deninger's papers studies L-functions and their special values. A classical example of an L-function is the Riemann zeta function ζ(s), for which formulas
Christopher_Deninger
Unproved conjecture in mathematics
prime p {\displaystyle p} . This L {\displaystyle L} -function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Galois group of a local or global field, introduced by Emil Artin as an expression appearing in the functional equation of an Artin L-function. Suppose that
Artin_conductor
Mathematical software
essential tool to calculate the higher-order QCD beta function. The mathematical structure of multiple zeta values has been researched with dedicated FORM programs
FORM (symbolic manipulation system)
FORM_(symbolic_manipulation_system)
Finite extension of the rationals
equation for the zeta-function are needed to define the function for all s). The Dedekind zeta-function generalizes the Riemann zeta-function in that ζ Q {\displaystyle
Algebraic_number_field
Type of shift space studied in ergodic theory
Artin–Mazur zeta function is defined as the formal power series ζ ( z ) = exp ( ∑ n = 1 ∞ | Fix ( T n ) | z n n ) , {\displaystyle \zeta (z)=\exp \left(\sum
Subshift_of_finite_type
Seventh letter in the Greek alphabet
Greek dialects to represent the voiceless glottal fricative, [h]. In this function, it was borrowed in the 8th century BC by the Etruscan and other Old Italic
Eta
Mathematical conjectures in class field theory
Weil group of F {\displaystyle F} there is an L-function L ( s , ρ ) {\displaystyle L(s,\rho )} and a local ε-factor ε ( s , ρ , ψ ) {\displaystyle \varepsilon
Local_Langlands_conjectures
Conjectures connecting number theory and geometry
of the Riemann zeta function) constructed from Hecke characters. The precise correspondence between these different kinds of L-functions constitutes Artin's
Langlands_program
sorority was Omicron Zeta Tau (Kappa Kappa Gamma) followed soon after by Sigma Beta Chi. By 1969 three local sororities and 9 local fraternities could be
Clemson University fraternities and sororities
Clemson_University_fraternities_and_sororities
Mathematical function with multiple real-number arguments
{\begin{aligned}&\zeta :\Xi \to \mathbb {R} ,\\&\zeta =\zeta (\xi _{1},\xi _{2},\ldots ,\xi _{m}),\end{aligned}}} is a function composition defined on X, in other terms
Function of several real variables
Function_of_several_real_variables
Mathematical theorem
let ζ m {\displaystyle \zeta _{m}} be a primitive mth root of unity, and let L = Q ( ζ m ) {\displaystyle L=\mathbb {Q} (\zeta _{m})} be the mth cyclotomic
Artin_reciprocity
Diocese of the Serbian Orthodox Church
by Saint Sava on which occasion the Eparchy of Zeta was established. The seat of the bishops of Zeta was the Monastery of Holy Archangel Michael in Prevlaka
Metropolitanate of Montenegro and the Littoral
Metropolitanate_of_Montenegro_and_the_Littoral
Property of a dynamical system where solutions near an equilibrium point remain so
{\displaystyle {\dot {\zeta }}(t)=g{\big (}t,\zeta (t){\big )};t\geq t_{0},} ζ ( t 0 ) = ζ 0 , {\displaystyle \zeta (t_{0})=\zeta _{0},} where ( t 0 , ζ
Lyapunov_stability
Differential equations involving stochastic processes
) t < ζ {\displaystyle (X_{t})_{t<\zeta }} up to life time ζ {\displaystyle \zeta } , s.t. for each test function f ∈ C c ∞ ( M ) {\displaystyle f\in
Stochastic differential equation
Stochastic_differential_equation
DC Comics fictional character
repeatedly traveling to a planet in the Alpha Centauri star system by using a "Zeta-beam" altered by space radiation. Since Adam Strange was the first human
Adam_Strange
Fourteenth letter in the Greek alphabet
distribution The symmetric function equation of the Riemann zeta function in mathematics, also known as the Riemann xi function A universal set in set theory
Xi_(letter)
travel, tourism, insurance
LOCAL ZETA-FUNCTION
LOCAL ZETA-FUNCTION
Girl/Female
Greek Native American
Stone; rock.
Girl/Female
Muslim
Female
Hebrew
(× Ö¶×˜Ö·×¢) Hebrew unisex name NETA means meaning "plant, shrub."
Female
Greek
(ΖÎνα) Contracted form of Greek Zenia, ZENA means "stranger, foreigner," but sometimes rendered "hospitable (esp. to foreigners)."
Biblical
watch-tower, associated with modern Zeita|Wadi Zeita
Female
Native American
 Native American Blackfoot name PETA means "golden eagle." Compare with another form of Peta.
Female
Polish
Feminine form of Polish Józef, JÓZEFA means "(God) shall add (another son)."Â
Female
Italian
Italian name ZITA means "little girl."Â
Girl/Female
Indian
Love
Male
French
French Provençal form of Latin Benedictus, BÉNÉZET means "blessed."Â
Female
Italian
 Variant spelling of Italian Zita, ZETA means "little girl." Compare with another form of Zeta.
Female
Spanish
 Short form of Spanish Aleta, LETA means "winged." Compare with another form of Leta.
Boy/Male
American, Australian, British, English, French
Faithful; True
Boy/Male
English American French
Faithful; unswerving.
Girl/Female
Greek American
Speaker.
Female
Persian/Iranian
 Short form of Persian Zenana, ZENA means "woman." Compare with another form of Zena.
Girl/Female
Muslim
Pretty
Female
English
English name derived from the second letter of the Greek alphabet, beta, related to Hebrew bet, BETA means "house."Â
Female
German
Short form of German Margarete, META means "pearl."
Girl/Female
Greek
Born last.
LOCAL ZETA-FUNCTION
LOCAL ZETA-FUNCTION
LOCAL ZETA-FUNCTION
LOCAL ZETA-FUNCTION
LOCAL ZETA-FUNCTION
LOCAL ZETA-FUNCTION
LOCAL ZETA-FUNCTION
n.
A train which receives and deposits passengers or freight along the line of the road; a train for the accommodation of a certain district.
pl.
of Seta
n.
A Greek letter corresponding to our z.
n.
The common beet (Beta vulgaris).
n.
A principle, practice, form of speech, or other thing of local use, or limited to a locality.
a.
Of or pertaining to a particular place, or to a definite region or portion of space; restricted to one place or region; as, a local custom.
a.
Loyal.
n.
A local European measure of length. See Canna.
a.
Uttered or modulated by the voice; oral; as, vocal melody; vocal prayer.
a.
Confined to no zone or region; not local.
n.
A small, short hair or bristle; a small seta.
n.
A vocal sound; specifically, a purely vocal element of speech, unmodified except by resonance; a vowel or a diphthong; a tonic element; a tonic; -- distinguished from a subvocal, and a nonvocal.
n.
On newspaper cant, an item of news relating to the place where the paper is published.
a.
Belonging to,or concerning, a focus; as, a focal point.
n.
A genus of large grasses of which the Indian corn (Zea Mays) is the only species known. Its origin is not yet ascertained. See Maize.
a.
Faithful; loyal; true.
n.
A local name of the burbot.
a.
Faithful; loyal.
a.
Alt. of Loral
n.
A district or local division, as of a province.
travel, tourism, insurance