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Generalizations of the Riemann zeta function
In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by ζ ( s 1 , … , s k ) = ∑ n 1 > n 2 > ⋯ > n k >
Multiple_zeta_function
Index of lists with the same name
Laplacian Motivic zeta function of a motive Multiple zeta function, or Mordell–Tornheim zeta function of several variables p-adic zeta function of a p-adic
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
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
Generalization of the Euler gamma function and the Barnes G-function
}{\partial s}}\zeta _{N}(s,w\mid a_{1},\ldots ,a_{N})\right|_{s=0}\right)\ ,} where ζ N {\displaystyle \zeta _{N}} is the Barnes zeta function. (This differs
Multiple_gamma_function
Arakawa–Kaneko zeta function is a generalisation of the Riemann zeta function which generates special values of the polylogarithm function. The zeta function ξ k
Arakawa–Kaneko_zeta_function
Special mathematical function
polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special
Polylogarithm
Identity obeyed by many special functions related to the gamma function
{\displaystyle k^{s}\zeta (s)=\sum _{n=1}^{k}\zeta \left(s,{\frac {n}{k}}\right),} where ζ ( s ) {\displaystyle \zeta (s)} is the Riemann zeta function. This is a
Multiplication_theorem
Function in analytic number theory
expansion of the Riemann zeta function, ζ(s) — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ζ*(s)
Dirichlet_eta_function
Transcendental single-variable function
tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function. The Clausen function of order 2 – often referred
Clausen_function
Barnes zeta function is a generalization of the Riemann zeta function introduced by E. W. Barnes (1901). It is further generalized by the Shintani zeta function
Barnes_zeta_function
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
Extension of superfactorials to the complex numbers
{\zeta (k)}{k+1}}z^{k+1}.} It is valid for 0 < z < 1 {\displaystyle \,0<z<1} . Here, ζ ( x ) {\displaystyle \,\zeta (x)} is the Riemann zeta function:
Barnes_G-function
rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or
Rational_zeta_series
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)
Evaluates the Riemann zeta function at many points
Odlyzko–Schönhage algorithm is a fast algorithm for evaluating the Riemann zeta function at many points, introduced by (Odlyzko & Schönhage 1988). The main point
Odlyzko–Schönhage_algorithm
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
Summatory function of the Möbius function
3+M\left({\frac {x}{4}}\right)\log 4+\cdots .} Assuming that the Riemann zeta function has no multiple non-trivial zeros, one has the "exact formula" by the residue
Mertens_function
In mathematics, the Jacobi zeta function Z(u) is the logarithmic derivative of the Jacobi theta function Θ(u). It is also commonly denoted as zn ( u
Jacobi_zeta_function
Concept in mathematics
{\bigl [}\zeta '(-1,z)-\zeta '(-1){\bigr ]}} where ζ′(z) denotes the derivative of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta function and ζ
K-function
Infinite products of functions indexed by primes
{\zeta (s)^{2}}{\zeta (2s)}}.} Since for even values of s the Riemann zeta function ζ(s) has an analytic expression in terms of a rational multiple of
Euler_product
Axiomatic definition of a class of L-functions
the essential properties satisfied by most functions that are commonly called L-functions or zeta functions. Although the exact nature of the class is
Selberg_class
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
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
Japanese mathematician
Tsumura (2011). "Shuffle products for multiple zeta values and partial fraction decompositions of zeta-functions of root systems". Mathematische Zeitschrift
Kohji_Matsumoto
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 function
\zeta (j)\ a_{n-j}}\ }{n-1}}={\frac {\ \gamma \ a_{n-1}-\zeta (2)\ a_{n-2}+\zeta (3)\ a_{n-3}-\cdots \ }{n-1}}} where ζ is the Riemann zeta function.
Reciprocal_gamma_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
Field of combinatorics using complex analysis
is an admissible function, then [ z n ] F ( z ) ∼ F ( ζ ) ζ n + 1 2 π f ″ ( ζ ) {\displaystyle [z^{n}]F(z)\sim {\frac {F(\zeta )}{\zeta ^{n+1}{\sqrt {2\pi
Analytic_combinatorics
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
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
International collegiate fraternity
Zeta Psi (ΖΨ) is an international collegiate fraternity. It was founded in 1847 at New York University. The fraternity has over 100 chapters, with roughly
Zeta_Psi
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
Type of mathematical functions
{\displaystyle \displaystyle f(z)=\int _{\partial D}f(\zeta )\omega (\zeta ,z).} Holomorphic functions of several complex variables satisfy an identity theorem
Function of several complex variables
Function_of_several_complex_variables
Japanese mathematician
number theory, especially analytic number theory, multiple trigonometric function theory, zeta functions and automorphic forms. He is currently a professor
Nobushige_Kurokawa
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
Function in q-analog theory
Yamasaki, Yoshinori (December 2006). "On q-Analogues of the Barnes Multiple Zeta Functions". Tokyo Journal of Mathematics. 29 (2): 413–427. arXiv:math/0412067
Q-gamma_function
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
'Best' approximation of a function by a rational function of given order
Riemann zeta function. Padé approximants can be used to extract critical points and exponents of functions. In thermodynamics, if a function f(x) behaves
Padé_approximant
Class of mathematical functions
'(w)=A\wp (w)+B.} Then the elliptic function ℘ ′ ( ζ ) − A ℘ ( ζ ) − B {\displaystyle \wp '(\zeta )-A\wp (\zeta )-B} has a pole of order three at zero
Weierstrass_elliptic_function
Branch of functional analysis
\nolimits _{\Gamma }{\frac {f(\zeta )}{\zeta -z}}\,d\zeta } for any z in U. The idea is to extend this formula to functions taking values in the Banach space
Holomorphic functional calculus
Holomorphic_functional_calculus
expressing the leading coefficient of the Taylor series for the Dedekind zeta function of a number field as the product of a regulator related to S-units of
Stark_conjectures
Function studied by Ramanujan
In mathematics, the Ramanujan tau function, studied by Srinivasa Ramanujan, is the function τ : N → Z {\displaystyle \tau :\mathbb {N} \to \mathbb {Z}
Ramanujan_tau_function
Set of functions used to represent the electronic wave function
multiple basis functions corresponding to each valence atomic orbital are called valence double, triple, quadruple-zeta, and so on, basis sets (zeta,
Basis_set_(chemistry)
Mathematical procedure
value of π. PSLQ has also helped find new identities involving multiple zeta functions and their appearance in quantum field theory; and in identifying
Integer_relation_algorithm
Approximate identity involving logarithms of primes
{\zeta ^{\prime \prime }(s)}{\zeta (s)}}=\left({\frac {\zeta ^{\prime }(s)}{\zeta (s)}}\right)^{\prime }+\left({\frac {\zeta ^{\prime }(s)}{\zeta (s)}}\right)^{2}=\sum
Selberg's_identity
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
American mathematician
the zeros of the Riemann zeta function, is known for his development of large sieve methods, and is the author of multiple books on number theory and
Hugh_Lowell_Montgomery
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
Integral criterion for holomorphy
define the function F to be F ( z ) = ∫ γ f ( ζ ) d ζ . {\displaystyle F(z)=\int _{\gamma }f(\zeta )\,d\zeta .} To see that the function is well-defined
Morera's_theorem
C library for arbitrary-precision floating-point arithmetic
exp(x)−1 functions (log1p and expm1), the six trigonometric and hyperbolic functions and their inverses, the gamma, zeta and error functions, the arithmetic–geometric
GNU_MPFR
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
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
Natural number
sixth floor of imaginary parts of non-trivial zeroes in the Riemann zeta function. It is in equivalence with the sum of ceilings of the first two such
37_(number)
Mathematical concept in number theory
restricting the product to primes up to a specified bound. For the Riemann zeta function, it takes the form P ( s , x ) = ∏ p ≤ x ( 1 − 1 p s ) − 1 , {\displaystyle
Partial_Euler_Product
Physical characteristic of oscillating systems
example the transfer function is H ( s ) = 2 ζ ω 0 s s 2 + 2 ζ ω 0 s + ω 0 2 . {\displaystyle H(s)={\frac {2\zeta \omega _{0}s}{s^{2}+2\zeta \omega _{0}s+\omega
Resonance
German mathematician and computer scientist
A. M.; Schönhage, A. (1988). "Fast Algorithms for Multiple Evaluations of the Riemann Zeta Function". Transactions of the American Mathematical Society
Arnold_Schönhage
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
Multivalued function in mathematics
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse
Lambert_W_function
Hashing technique
the server in which we can place the BLOB: ζ = β % n {\displaystyle \zeta =\beta \ \%\ n} ; hence the BLOB will be placed in the server whose server
Consistent_hashing
finite fields Riemann theta function Riemann Xi function Riemann zeta function Riemann–Siegel formula Riemann–Siegel theta function Free Riemann gas also called
List of things named after Bernhard Riemann
List_of_things_named_after_Bernhard_Riemann
Function in number theory given by Srinivasa Ramanujan
Euler's totient function. μ ( n ) {\displaystyle \mu (n)} is the Möbius function. ζ ( s ) {\displaystyle \zeta (s)} is the Riemann zeta function. The following
Ramanujan's_sum
Identity in analytic number theory
the expansion of the logarithmic derivative of the Riemann zeta function in terms of functions which are partial Dirichlet series respectively truncated
Vaughan's_identity
to multiple sororities. Originally an educational sorority Originally affiliated with Alpha Sigma Alpha. Merged with Lambda Omega (see Delta Zeta). Originally
List of social sororities and women's fraternities
List_of_social_sororities_and_women's_fraternities
Protein-coding gene in the species Homo sapiens
T-cell surface glycoprotein CD3 zeta chain also known as T-cell receptor T3 zeta chain or CD247 (Cluster of Differentiation 247) is a protein that in humans
T-cell surface glycoprotein CD3 zeta chain
T-cell_surface_glycoprotein_CD3_zeta_chain
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)
Mathematical function
(November 2002). "Vinogradov's Integral and Bounds for the Riemann Zeta Function" (PDF). Proc. London Math. Soc. 85 (3): 565–633. arXiv:1910.08209. doi:10
Landau's_function
Theorem about polynomials
a_{0}+a_{1}\zeta +a_{2}\zeta ^{2}+\cdots +a_{n}\zeta ^{n}=0} which can be put as ∑ r = 0 n a r ζ r = 0. {\displaystyle \sum _{r=0}^{n}a_{r}\zeta ^{r}=0.}
Complex conjugate root theorem
Complex_conjugate_root_theorem
Every polynomial has a real or complex root
{\displaystyle |\zeta |^{n}\leq \|a\|_{p}\left(|\zeta |^{q(n-1)}+\cdots +|\zeta |^{q}+1\right)^{\frac {1}{q}}=\|a\|_{p}\left({\frac {|\zeta |^{qn}-1}{|\zeta |^{q}-1}}\right)^{\frac
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Fermat's theorem on sums of two squares Riemann zeta function Basel problem on ζ(2) Hurwitz zeta function Bernoulli number Agoh–Giuga conjecture Von Staudt–Clausen
List_of_number_theory_topics
Function in thermodynamics and statistical physics
principle, the total partition function must be divided by a N! (N factorial): Z = ζ N N ! . {\displaystyle Z={\frac {\zeta ^{N}}{N!}}.} This is to ensure
Partition function (statistical mechanics)
Partition_function_(statistical_mechanics)
Certain type of divisor of an integer
function ζ ( s ) ζ ( s − k ) ( 1 − 2 k − s ) ζ ( 2 s − k ) ( 1 − 2 k − 2 s ) = ∑ n ≥ 1 σ k ( o ) ∗ ( n ) n s . {\displaystyle {\frac {\zeta (s)\zeta
Unitary_divisor
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
2nd century AD trigonometric table
for arcs that were multiples of 7+1/2° = π/24 radians). Since the 8th and 9th centuries, the sine and other trigonometric functions have been used in
Ptolemy's_table_of_chords
Formal power series
(a_{n};s)\zeta (s)=\operatorname {DG} (b_{n};s),} where ζ(s) is the Riemann zeta function. The sequence ak generated by a Dirichlet series generating function (DGF)
Generating_function
American mathematician (born 1945)
for the interesting factor of their zeta functions. For exponential sums, they expressed the degree of the L-function (or its reciprocal) given in terms
Steven_Sperber
Fundamental trigonometric functions
equation for the Riemann zeta-function, ζ ( s ) = 2 ( 2 π ) s − 1 Γ ( 1 − s ) sin ( π 2 s ) ζ ( 1 − s ) . {\displaystyle \zeta (s)=2(2\pi )^{s-1}\Gamma
Sine_and_cosine
Mathematical functions
_{2n}{\frac {(2\pi )^{2n}}{(2n)!}}=2\zeta (2n),\quad n\geq 1} where ζ {\displaystyle \zeta } is the Riemann zeta function. The Hurwitz numbers H n , {\displaystyle
Lemniscate_elliptic_functions
Rules for computing derivatives of functions
rules for computing the derivative of a function in calculus. Unless otherwise stated, all functions are functions of real numbers ( R {\textstyle \mathbb
Differentiation_rules
Complex-valued arithmetic function
Euler's totient function. ζ n {\displaystyle \zeta _{n}} is a complex primitive n-th root of unity: ζ n n = 1 , {\displaystyle \zeta _{n}^{n}=1,} but
Dirichlet_character
Mathematical theorem
x+\sum _{k=2}^{\infty }{\frac {\,\zeta (k)\,}{k}}\,(-x)^{k}} where ζ ( k ) {\textstyle \zeta (k)} is the Riemann zeta function. Then applying Ramanujan master
Ramanujan's_master_theorem
Largest integer that divides given integers
probability 1/ζ(k) as n goes to infinity, where ζ refers to the Riemann zeta function. (See coprime for a derivation.) This result was extended in 1987 to
Greatest_common_divisor
Protein family
protein domain Zeta (ζ) toxin refers to a protein domain found in prokaryotes, which acts as a UDP-N-acetylglucosamine kinase. Its function is to inhibit
Zeta_toxin_protein_domain
Rational number sequence
Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent
Bernoulli_number
Function whose domain is the positive integers
generating function of the Möbius function is the inverse of the zeta function: ζ ( s ) ∑ n = 1 ∞ μ ( n ) n s = 1 , ℜ s > 1. {\displaystyle \zeta (s)\,\sum
Arithmetic_function
Mathematical concept
integrals such as the Gamma function, the Riemann zeta function, and other Hankel functions (which are Bessel functions of the third kind). The Hankel
Hankel_contour
Branch of pure mathematics
understood through the study of analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic
Number_theory
Divergent sum of positive unit fractions
function on all complex numbers except x = 1 {\displaystyle x=1} , where the extended function has a simple pole. Other important values of the zeta function
Harmonic_series_(mathematics)
American actor (born 1944)
Nations Messenger of Peace in 1998. He has been married to actress Catherine Zeta-Jones since 2000. Douglas was born in New Brunswick, New Jersey, the first
Michael_Douglas
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
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
Mathematical problems related to differential equations
ξ d η . {\displaystyle d\zeta \wedge d{\bar {\zeta }}=(d\xi +id\eta )\wedge (d\xi -id\eta )=-2id\xi d\eta .} If a function M ( z ) {\displaystyle M(z)}
Riemann–Hilbert_problem
Number divisible only by 1 and itself
the zeros of the Riemann zeta function ζ ( s ) {\displaystyle \zeta (s)} are located. This function is an analytic function on the complex numbers. For
Prime_number
Australian mathematician and computer scientist
1979 he showed that the first 75 million complex zeros of the Riemann zeta function lie on the critical line, providing some experimental evidence for the
Richard_P._Brent
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
Mathematical function, inverse of an exponential function
logarithm by Li1 (z) = −ln(1 − z). Moreover, Lis (1) equals the Riemann zeta function ζ(s). Mathematics portal Arithmetic portal Chemistry portal Geography
Logarithm
Mathematical theorem
^{n}}{\frac {u(x+h\nu )-u(x)}{h}}\zeta (x)\,d{\mathcal {L}}^{n}(x)=-\int _{\mathbf {R} ^{n}}{\frac {\zeta (x)-\zeta (x-h\nu )}{h}}u(x)\,d{\mathcal {L}}^{n}(x)
Rademacher's_theorem
Abstract mathematical group
{\big [}(\xi ,\alpha ),(\zeta ,\beta ){\big ]}={\bigg (}[\xi ,\zeta ],-{\frac {1}{24\pi }}\int _{0}^{2\pi }{\text{d}}x\,\xi (x)\zeta '''(x){\bigg )}} where
Virasoro_group
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
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
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MULTIPLE ZETA-FUNCTION
MULTIPLE ZETA-FUNCTION
MULTIPLE ZETA-FUNCTION
MULTIPLE ZETA-FUNCTION
MULTIPLE ZETA-FUNCTION
MULTIPLE ZETA-FUNCTION
MULTIPLE ZETA-FUNCTION
MULTIPLE ZETA-FUNCTION
MULTIPLE ZETA-FUNCTION
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