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Type of zeta function
mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes
Arithmetic_zeta_function
{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
List_of_zeta_functions
Generalization of the Riemann zeta function for algebraic number fields
functional equation. Values of Dedekind zeta functions encode important arithmetic data of K. The Dedekind zeta function is named for Richard Dedekind, who
Dedekind_zeta_function
Function whose domain is the positive integers
\log _{e}(x)} . In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive
Arithmetic_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
Analytic function in mathematics
The Riemann zeta function or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a mathematical function of a complex variable
Riemann_zeta_function
Mathematical function associated to algebraic varieties
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
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
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
P-adic_L-function
Function on an integer n which is log(p) if n equals p^k and zero otherwise
Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that
Von_Mangoldt_function
Arithmetic function related to the divisors of an integer
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
Divisor_function
Number of integers coprime to and less than n
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 )
Euler's_totient_function
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
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
Arithmetic function
Liouville function, named after French mathematician Joseph Liouville and denoted λ ( n ) {\displaystyle \lambda (n)} , is an important arithmetic function. Its
Liouville_function
Function representing the number of primes less than or equal to a given number
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
Prime-counting_function
Function equal to the product of its values on coprime factors
In number theory, a multiplicative function is an arithmetic function f {\displaystyle f} of a positive integer n {\displaystyle n} with the property that
Multiplicative_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
Special function of two variables
the function by a factor of ζ ( 2 s ) {\displaystyle \zeta (2s)} , where ζ {\displaystyle \zeta } is the Riemann zeta function. Viewed as a function of
Real analytic Eisenstein series
Real_analytic_Eisenstein_series
Multiplicative function in number theory
partition function is the Riemann zeta function. This idea underlies Alain Connes's attempted proof of the Riemann hypothesis. The Möbius function is multiplicative
Möbius_function
Relation between pairs of arithmetic functions
{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
Möbius_inversion_formula
Arithmetic function
{\frac {1}{2\zeta (2)}}n^{2}+O\left(n\log n\right)={\frac {3}{\pi ^{2}}}n^{2}+O\left(n\log n\right),} where ζ(2) is the Riemann zeta function evaluated at
Totient_summatory_function
arithmetic function is some simpler or better-understood function which takes the same values "on average". Let f {\displaystyle f} be an arithmetic function
Average order of an arithmetic function
Average_order_of_an_arithmetic_function
Mathematical series
definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. Specifically, the Riemann zeta function ζ(s) is the Dirichlet
Dirichlet_series
higher-dimensional generalization of the Goss zeta function. Goss, David (1996), Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer
Goss_zeta_function
Russian mathematician
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
Ivan_Fesenko
Special mathematical function
Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular
Dirichlet_beta_function
Mathematical conjecture about zeros of L-functions
zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Exploring properties of the integers with complex analysis
results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's
Analytic_number_theory
Use of a Dirichlet series expansion to calculate the complex function
Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas (Various Observations
Proof of the Euler product formula for the Riemann zeta function
Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function
Field arising from a quotient ring by a maximal ideal
transcendence degree of the residue field of the generic point. Arithmetic zeta function Dummit, D. S.; Foote, R. (2004). Abstract Algebra (3 ed.). Wiley
Residue_field
German mathematician (born 1958)
University of Münster. Deninger's research focuses on arithmetic geometry, including applications to L-functions. Deninger obtained his doctorate from the University
Christopher_Deninger
J. Tate, Algebraic cycles and poles of zeta functions in the volume (O. F. G. Schilling, editor), Arithmetical Algebraic Geometry, pages 93–110 (1965)
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Arithmetical function
Dirichlet generating function of μ {\displaystyle \mu } is 1 / ζ ( s ) {\displaystyle 1/\zeta (s)} and the Dirichlet generating function of n k {\displaystyle
Jordan's_totient_function
Mathematical theory
arithmetic surfaces attached to smooth projective curves over number fields, with the aim of proving certain results, known in the case of function fields
Arakelov_theory
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
Arithmetic of abelian varieties
Arithmetic_of_abelian_varieties
Characterization of how many integers are prime
particular, the Riemann zeta function). The first such distribution found is π(N) ~ N/log(N), where π(N) is the prime-counting function (the number of primes
Prime_number_theorem
French mathematician
Leichtnam, Eric (2005), "An invitation to Deninger's work on arithmetic zeta functions", Geometry, spectral theory, groups, and dynamics, Contemp. Math
Éric_Leichtnam
Theorem on the number of primes in arithmetic sequences
related the study of prime numbers to what is known now as the Riemann zeta function: he showed that the value ζ(1) reduces to a ratio of two infinite products
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
Constants of the mathematical zeta function
Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted ζ ( s ) {\displaystyle \zeta (s)}
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
Branch of algebraic geometry
Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined
Arithmetic_geometry
Type of mathematical function
L-function of the principal character χ 0 {\displaystyle \chi _{0}} modulo q {\displaystyle q} can be expressed in terms of the Riemann zeta function:
Dirichlet_L-function
Synchrotron function Riemann zeta function: A special case of Dirichlet series. Riemann Xi function Dirichlet eta function: An allied function. Dirichlet
List of mathematical functions
List_of_mathematical_functions
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
following outline is provided as an overview of and topical guide to arithmetic: Arithmetic is an elementary branch of mathematics that deals with numerical
Outline_of_arithmetic
Analytic function that does not satisfy a polynomial equation
logarithm and inverse trigonometric functions. All special functions such as the gamma, error, bessel, and Riemann zeta functions are transcendental. Equations
Transcendental_function
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
Mathematical function
u} is 2 K ( m ) {\displaystyle 2K(m)} . It is related to the Jacobi zeta function by Z ( φ | m ) = zn ( F ( φ | m ) | m ) . {\displaystyle Z(\varphi
Jacobi_elliptic_functions
Nearest integers from a number
Zeta-function (2nd ed.), Oxford: Oxford U. P., ISBN 0-19-853369-1 Wikimedia Commons has media related to Floor and ceiling functions. "Floor function"
Floor_and_ceiling_functions
Mathematical function
expression for ψ(x) as a sum over the nontrivial zeros of the Riemann zeta function: ψ 0 ( x ) = x − ∑ ρ x ρ ρ − ζ ′ ( 0 ) ζ ( 0 ) − 1 2 log ( 1 − x −
Chebyshev_function
Branch of pure mathematics
of mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of
Number_theory
Class of functions behaving "like" periodic functions
is a "simpler" function than f {\displaystyle f} . What it means to be "simpler" is vague. A simple case (sometimes called arithmetic quasiperiodic) is
Quasiperiodic_function
Elementary functions and their finitely iterated integrals
directly from the definition that the set of Liouvillian functions is closed under arithmetic operations, composition, and integration. It is also closed
Liouvillian_function
Number divisible only by 1 and itself
derived from the fundamental theorem of arithmetic, and shows the close connection between the zeta function and the prime numbers. It leads to another
Prime_number
Seven mathematical problems with a US$1 million prize for each solution
problem was given by Stephen Cook. The Riemann zeta function ζ ( s ) {\displaystyle \zeta (s)} is a function whose arguments may be any complex number other
Millennium_Prize_Problems
Number of prime factors of a natural number
counts the total number of prime factors with multiplicity (see arithmetic function). That is, if we have a prime factorization of n {\displaystyle n}
Prime_omega_function
Summatory function of the Möbius function
Riemann zeta function. Using the Euler product, one finds that 1 ζ ( s ) = ∏ p ( 1 − p − s ) = ∑ n = 1 ∞ μ ( n ) n s , {\displaystyle {\frac {1}{\zeta (s)}}=\prod
Mertens_function
Point not touching any other point
only finitely many points. This makes it possible to define the arithmetic zeta function of such a scheme. Let X {\displaystyle X} be affine scheme (or
Closed_point
Summatory function of the divisor-counting function
summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function
Divisor_summatory_function
Algebraic curve in mathematics
variable, L, the Hasse–Weil zeta function of E over Q. This function is a variant of the Riemann zeta function and Dirichlet L-functions. It is defined as an
Elliptic_curve
Mathematical function
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
Riesz_function
Branch of mathematics
of arithmetic functions and zeta functions is extensive. The idea is to extend the various arguments and techniques of arithmetic functions and zeta functions
Abstract analytic number theory
Abstract_analytic_number_theory
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
Calculations where numbers' precision is only limited by computer memory
arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations
Arbitrary-precision arithmetic
Arbitrary-precision_arithmetic
Arithmetical function
(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-1)}{\zeta (2s)}}.} This is also a consequence of the fact that we can write the function as a Dirichlet convolution
Dedekind_psi_function
Symbolic description of a mathematical object
exponents. An unknown number was called ζ {\displaystyle \zeta } . The square of ζ {\displaystyle \zeta } was Δ v {\displaystyle \Delta ^{v}} ; the cube was
Expression_(mathematics)
Polynomial sequence
functions, and with the Euler–MacLaurin formula. These polynomials occur in the study of many special functions and, in particular, the Riemann zeta function
Bernoulli_polynomials
Mathematical operation on arithmetical functions
convolution (or divisor convolution) is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav
Dirichlet_convolution
English-French mathematician
Motives. He also works on Zeta functions in quantum field theory. He was elected a Fellow of the Royal Society in 2026. Multiple zeta values and periods of
Francis_Brown_(mathematician)
Set-theoretic function
function ψ {\displaystyle \psi } remains "stuck" at ζ 0 {\displaystyle \zeta _{0}} for some time: ψ ( α ) = ζ 0 {\displaystyle \psi (\alpha )=\zeta _{0}}
Ordinal_collapsing_function
Rational number sequence
to the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties. For example
Bernoulli_number
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
Nowhere analytic, infinitely differentiable function
(2017). "Arithmetic of the Fabius function". arXiv:1702.06487 [math.NT]. Sloane, N. J. A. (ed.). "Sequence A272755 (Numerators of the Fabius function F(1/2^n))"
Fabius_function
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
Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
Riemann zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural logarithm function and
Harmonic_number
Japanese mathematician
the Ihara zeta function. Ihara received his PhD at the University of Tokyo in 1967 with thesis Hecke polynomials as congruence zeta functions in elliptic
Yasutaka_Ihara
on arithmetic progressions Linnik's theorem Elliott–Halberstam conjecture Functional equation (L-function) Chebotarev's density theorem Local zeta function
List_of_number_theory_topics
Mathematical concept
congruence relation, implies that the Hasse–Weil zeta function of a modular curve is a product of L-functions associated to explicitly determined modular forms
Shimura_variety
American mathematician
function, by studying arithmetic hyperbolic 3-manifolds. He later formulated a general conjecture giving formulas for special values of Dedekind zeta
Don_Zagier
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
Natural number
of imaginary parts of non-trivial zeroes in the Riemann zeta function, ζ . {\displaystyle \zeta .} 29 is the largest prime factor of the smallest number
29_(number)
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
Mathematical functions that quantify complexity
approximation, Diophantine equations, arithmetic geometry, and mathematical logic. An early form of height function was proposed by Giambattista Benedetti
Height_function
Expression for sums of powers
This result agrees with the value of the Riemann zeta function ζ ( s ) = ∑ n = 1 ∞ 1 n s {\textstyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}} for
Faulhaber's_formula
Number, approximately 3.14
{\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s).} Furthermore, the derivative of the zeta function satisfies
Pi
Difference between logarithm and harmonic series
} . Evaluations of the digamma function at rational values. The Laurent series expansion for the Riemann zeta function*, where it is the first of the
Euler's_constant
related to an algebraic number field, whose partition function is related to the Dedekind zeta function of the number field. Bost & Connes (1995) introduced
Bost–Connes_system
Operation on formal power series
other series for the zeta-function-related cases of the Legendre chi function, the polygamma function, and the Riemann zeta function include χ 1 ( z ) =
Generating function transformation
Generating_function_transformation
Associative algebra used in combinatorics
of the zeta function is the Möbius function μ(a, b); every value of μ(a, b) is an integral multiple of 1 in the base ring. The Möbius function can also
Incidence_algebra
Mathematical conjecture
group (its number of elements) to the value of the Dedekind zeta function ζ F {\displaystyle \zeta _{F}} . More specifically, let F be a totally real number
Birch–Tate_conjecture
Uses of the constant
_{2}=\zeta (z+\omega _{2};\Omega )-\zeta (z;\Omega )} where ζ {\displaystyle \zeta } is the Weierstrass zeta function ( η 1 {\displaystyle \eta _{1}} and
List_of_formulae_involving_π
Potential counterexample to the generalized Riemann hypothesis
character. For an integer q ≥ 1, a Dirichlet character modulo q is an arithmetic function χ : Z → C {\textstyle \chi \colon \mathbb {Z} \to \mathbb {C} } satisfying
Siegel_zero
On the distribution of prime numbers
different problems: the original Riemann hypothesis for the Riemann zeta function the solvability of two-variable, linear, diophantine equations in prime
Hilbert's_eighth_problem
Rational numbers with root 5 added
the cyclotomic field Q ( ζ ) {\displaystyle \mathbb {Q} (\zeta )} and the arithmetic of matrices spanned by I {\displaystyle \mathbf {I} } and
Golden_field
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
Formula for the sum of an arithmetic function
sum of an arithmetic function, by means of an inverse Mellin transform. Let { a ( n ) } {\displaystyle \{a(n)\}} be an arithmetic function, and let g
Perron's_formula
Sigmoid shape special function
D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow). Despite the name "imaginary error function", erfi(x) is real
Error_function
C library for arbitrary-precision floating-point arithmetic
functions (log1p and expm1), the six trigonometric and hyperbolic functions and their inverses, the gamma, zeta and error functions, the arithmetic–geometric
GNU_MPFR
Type of mathematical group
and ζ F {\displaystyle \zeta _{F}} its Dedekind zeta function. Let Γ O {\displaystyle \Gamma _{\mathcal {O}}} be the arithmetic group obtained from O {\displaystyle
Arithmetic_Fuchsian_group
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
ARITHMETIC ZETA-FUNCTION
ARITHMETIC ZETA-FUNCTION
Female
German
Short form of German Margarete, META means "pearl."
Girl/Female
Greek Native American
Stone; rock.
Girl/Female
Indian
Love
Female
Native American
 Native American Blackfoot name PETA means "golden eagle." Compare with another form of Peta.
Girl/Female
Muslim
Girl/Female
Greek American
Speaker.
Girl/Female
Greek
Born last.
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.
Female
Persian/Iranian
 Short form of Persian Zenana, ZENA means "woman." Compare with another form of Zena.
Girl/Female
Hebrew American Spanish
Grace.
Male
French
French Provençal form of Latin Benedictus, BÉNÉZET means "blessed."Â
Female
Hebrew
(× Ö¶×˜Ö·×¢) Hebrew unisex name NETA means meaning "plant, shrub."
Female
English
English name derived from the second letter of the Greek alphabet, beta, related to Hebrew bet, BETA means "house."Â
Girl/Female
Danish American Greek Persian Latin
Biblical
watch-tower, associated with modern Zeita|Wadi Zeita
Female
Italian
Italian name ZITA means "little girl."Â
Female
Polish
Feminine form of Polish Józef, JÓZEFA means "(God) shall add (another son)."Â
Girl/Female
Muslim
Pretty
Female
Greek
(ΖÎνα) Contracted form of Greek Zenia, ZENA means "stranger, foreigner," but sometimes rendered "hospitable (esp. to foreigners)."
ARITHMETIC ZETA-FUNCTION
ARITHMETIC ZETA-FUNCTION
Boy/Male
Hindu, Indian
The Bringer of Hope and Smiles; God's Gift
Girl/Female
Tamil
Treasure of water
Girl/Female
Tamil
Aleshanee | அலேஷாநீ
She plays all the time
Girl/Female
Arabic, Muslim
Honoured; Elevated
Boy/Male
Arabic, Muslim
Sign of Allah
Boy/Male
Muslim/Islamic
Guardian
Girl/Female
English Latin
Follower of Christ.
Boy/Male
Tamil
Sudhanshu | ஸà¯à®¤à®¾à®‚à®·à¯Â
The Moon
Girl/Female
Muslim
Walking gently
Girl/Female
Indian
Seasonal
ARITHMETIC ZETA-FUNCTION
ARITHMETIC ZETA-FUNCTION
ARITHMETIC ZETA-FUNCTION
ARITHMETIC ZETA-FUNCTION
ARITHMETIC ZETA-FUNCTION
a.
Of or pertaining to arithmetic; according to the rules or method of arithmetic.
n.
A small, short hair or bristle; a small seta.
v. i.
To use figures in a mathematical process; to do sums in arithmetic.
n.
Arithmetical subtraction.
n.
One of the spinelike feathers at the base of the bill of certain birds.
n.
A Greek letter corresponding to our z.
adv.
The arithmetical character 0; a cipher. See Cipher.
v. t.
To subject to arithmetical division.
n.
The common beet (Beta vulgaris).
n.
That part of arithmetic which treats of adding numbers.
n.
One skilled in arithmetic.
n.
The science of numbers; the art of computation by figures.
n.
A book containing the principles of this science.
n.
A system of arithmetic, in which numbers are expressed in a scale of 60; logistic arithmetic.
v. t.
To subtract by arithmetical operation; to deduct.
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.
pl.
of Seta
n.
Arithmetic.
a.
Sexagesimal, or made on the scale of 60; as, logistic, or sexagesimal, arithmetic.
adv.
Conformably to the principles or methods of arithmetic.