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The Selberg zeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zeta function ζ ( s ) = ∏ p ∈ P 1 1 − p − s {\displaystyle
Selberg_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
Index of lists with the same name
zeta function, alternative names for the Z function Ruelle zeta function Selberg zeta function of a Riemann surface Shimizu L-function Shintani zeta function
List_of_zeta_functions
Mathematical finite graph-associated function
mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate
Ihara_zeta_function
mathematics, the Selberg conjecture, named after Atle Selberg, is a theorem about the density of zeros of the Riemann zeta function ζ(1/2 + it). It is
Selberg's zeta function conjecture
Selberg's_zeta_function_conjecture
Mathematical theorem
by the analogy, Selberg introduced the Selberg zeta function of a Riemann surface, whose analytic properties are encoded by the Selberg trace formula.
Selberg_trace_formula
Type of curve in geometry
Audrey (2011). "Selberg zeta function". Zeta Functions of Graphs: A Stroll through the Garden. Cambridge: Cambridge University Press. Selberg, Atle (1956)
Prime_geodesic
Zeta-like functions approximate arbitrary holomorphic functions
universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate
Zeta_function_universality
Mathematical concept
harmonic analysis on adelic spaces. Selberg trace formula Selberg zeta function The original prime counting function can easily be recovered via π ( x
Explicit formulae for L-functions
Explicit_formulae_for_L-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
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
Norwegian mathematician (1917–2007)
numbers and the zeros of the zeta function. He generally worked alone. His only coauthor was Sarvadaman Chowla. Selberg was awarded the 1986 Wolf Prize
Atle_Selberg
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
Evaluates a certain product of values of the Gamma function at rational values
In mathematics, the Chowla–Selberg formula is the evaluation of a certain product of values of the gamma function at rational values in terms of values
Chowla–Selberg_formula
Identity obeyed by many special functions related to the gamma function
theorem for the gamma functions can be understood to be a special case, for the trivial Dirichlet character, of the Chowla–Selberg formula. Formally similar
Multiplication_theorem
Axiomatic definition of a class of L-functions
In mathematics, the Selberg class is an axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms
Selberg_class
Mathematical Theory
the Rankin–Selberg method, introduced by Rankin (1939) and Selberg (1940), also known as the theory of integral representations of L-functions, is a technique
Rankin–Selberg_method
Arithmetic function related to the divisors of an integer
_{b}(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-a)\zeta (s-b)\zeta (s-a-b)}{\zeta (2s-a-b)}},} which is a special case of the Rankin–Selberg convolution. A Lambert
Divisor_function
Statement in number theory
{1}{2}}s(s-1)\pi ^{-s/2}\Gamma \left({\frac {s}{2}}\right)\zeta (s)} where ζ is the Riemann zeta function. Consider the sequence λ n = 1 ( n − 1 ) ! d n d s n
Li's_criterion
Mathematical conjecture about zeros of L-functions
for modular forms). Artin's conjecture Artin L-function Dirichlet L-function Dedekind zeta function Selberg class Grand Riemann hypothesis Davenport, Harold
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Mathematical concept
Langlands Program are Rankin-Selberg products of representations of GL(m) and GL(n). The resulting Rankin-Selberg L-functions satisfy a number of analytic
Automorphic_L-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
283–317. doi:10.1007/bf01211614. S2CID 126338046. Selberg, A. (1942). "On the zeros of Riemann's zeta-function". SHR. Norske Vid. Akad. Oslo. 10: 1–59. Karatsuba
Hardy–Littlewood zeta function conjectures
Hardy–Littlewood_zeta_function_conjectures
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
Function equal to the product of its values on coprime factors
{\displaystyle n} . See Selberg (1977). It is known that the classes of semimultiplicative and Selberg multiplicative functions coincide. They both satisfy
Multiplicative_function
Approximate identity involving logarithms of primes
In number theory, Selberg's identity is an approximate identity involving logarithms of primes found by Atle Selberg. The identity forms the crucial starting
Selberg's_identity
Generalization of the Riemann zeta function for algebraic number fields
the Dedekind zeta function of an algebraic number field K, usually denoted ζ K ( s ) {\displaystyle \zeta _{K}(s)} , is an analytic function that represents
Dedekind_zeta_function
Theorem of the three geodesics Curve-shortening flow Selberg trace formula Selberg zeta function Zoll surface Besse, A.: "Manifolds all of whose geodesics
Closed_geodesic
British mathematician
discontinuous groups (Fuchsian groups), different zeta functions (for example those of Ruelle and Selberg, in particular those associated with certain groups
Samuel_James_Patterson
Mathematical concept
with the Selberg class. Furthermore, all L-functions over arbitrary number fields are widely thought to be instances of standard L-functions for the general
Standard_L-function
polynomial Jacopo Riccati: Riccati–Bessel function Bernhard Riemann: Riemann zeta function, Riemann xi function Olinde Rodrigues: Rodrigues formula Leonard
List of eponyms of special functions
List_of_eponyms_of_special_functions
automorphic L-functions, and hence should be part of the Selberg class. There are also conjectures concerning the values of these L-functions at integers
Motivic_L-function
Topics referred to by the same term
Grothendieck–Lefschetz trace formula, that may be interpreted as a Selberg trace formula. List of zeta functions List of fixed point theorems This disambiguation page
Trace_formula
Characterization of how many integers are prime
the Riemann zeta function on the line R e ( s ) = 1 {\displaystyle Re(s)=1} , where complex analysis must be used. In March 1948, Atle Selberg established
Prime_number_theorem
Linear operators with a common spectrum
formula, via the Selberg zeta function. Sunada noticed that the method of constructing number fields with the same Dedekind zeta function could be adapted
Isospectral
On the distribution of prime numbers
of Riemann's zeta function are on σ = 1/2". Advances in Mathematics. 13 (4): 383–436. doi:10.1016/0001-8708(74)90074-7. MR 0564081. Selberg, Atle (1942)
Hilbert's_eighth_problem
Symmetric holomorphic function
308-339, 1979. Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J. reine angew. Math. 227, 86-110, 1967. Modular lambda function at Fungrim
Modular_lambda_function
Mathematical conjecture about the Riemann zeta function
Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible
Hilbert–Pólya_conjecture
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
Expresses a Gauss sum using a product of values of the p-adic gamma function
product of values of the p-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Davenport relation
Gross–Koblitz_formula
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
Equation in Fourier analysis
functional equation for the Riemann zeta function. One important such use of Poisson summation concerns theta functions: periodic summations of Gaussians
Poisson_summation_formula
Conjectures connecting number theory and geometry
on semisimple Lie groups, and in technical terms the trace formula of Selberg and others. What was new in Langlands' work, besides technical depth, was
Langlands_program
Russian mathematician
Selberg trace formula, Journal of Soviet Mathematics, vol. 8, 1977, pp. 171–199 Spectral theory of automorphic functions, the Selberg zeta-function,
Alexei_Venkov
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
Theorem on the number of primes in arithmetic sequences
the Riemann zeta function to the distribution of primes. The theorem represents the beginning of rigorous analytic number theory. Atle Selberg gave an elementary
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
/ 4 {\displaystyle 1/4} . Selberg's orthogonality conjecture: generalization of Mertens' theorem for functions in Selberg class. Bombieri–Lang conjecture:
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Unsolved problem in mathematics
the tau function is not completely multiplicative, the sums cannot be written using geometric series like in the case of the Riemann zeta function or Dirichlet
Ramanujan–Petersson conjecture
Ramanujan–Petersson_conjecture
Special function defined by an integral
Wiley-Interscience. ISBN 0-471-83138-7. p. 298 Chowla, S.; Selberg, A. (1949). "On Epstein's Zeta Function (I)". Proceedings of the National Academy of Sciences
Elliptic_integral
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
pages, for a total of 890 pages. 1983 – Selberg trace formula. Hejhal's proof of a general form of the Selberg trace formula consisted of 2 volumes with
List of long mathematical proofs
List_of_long_mathematical_proofs
Mathematical conjectures in class field theory
{\displaystyle \pi '} of general linear groups, there are local Rankin–Selberg convolution L-functions L ( s , π × π ′ ) {\displaystyle L(s,\pi \times \pi ')} and
Local_Langlands_conjectures
Existence of geodesic circles on surfaces
encoded analytically by the Selberg zeta function. The growth rate of the number of simple closed geodesics, as a function of their length, was investigated
Theorem of the three geodesics
Theorem_of_the_three_geodesics
theorem 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
American mathematician (born 1939)
Berlin-New York, 1973. vi+132 pp. Perry, Peter A.; Williams, Floyd L. Selberg zeta function and trace formula for the BTZ black hole. Int. J. Pure Appl. Math
Floyd_Williams
Conjecture in the representation theory of Lie groups
nonvanishing of the central value of the Rankin-Selberg L-functions II", Automorphic Representations, L-Functions and Applications: Progress and Prospects,
Gan–Gross–Prasad_conjecture
French mathematician
(1999). "A combinatorial proof of Bass's evaluations of the Ihara-Selberg zeta functions for graphs". Trans. Amer. Math. Soc. 351 (6): 2257–2274. arXiv:math/9806037
Dominique_Foata
Mahler's theorem Brun sieve Function field sieve General number field sieve Large sieve Larger sieve Quadratic sieve Selberg sieve Sieve of Atkin Sieve
List_of_number_theory_topics
American mathematician
since 1983. Hejhal works on analytic number theory, automorphic forms, the Selberg trace formula and quantum chaos. From 1972 to 1974 he was a Sloan Fellow
Dennis_Hejhal
Particular kind of exponential sum
{\displaystyle \mathbb {Q} \left(\zeta _{2^{\alpha -1}}+\zeta _{2^{\alpha -1}}^{-1}\right)} for 2α || m with α > 3. The Selberg identity: K ( a , b ; m ) =
Kloosterman_sum
Difference between two successive prime numbers
refers to the big O notation, ζ denotes the Riemann zeta function and π the prime-counting function. Knowing that any c > 1/6 is admissible, one obtains
Prime_gap
Type of generalization of periodic functions in Euclidean space
automorphic functions can be seen as generalizations of modular forms (as therefore elliptic curves), constructed by some zeta function analogue on an
Automorphic_form
Mathematical constant
generalized form of Newman's conjecture for L {\displaystyle L} -functions in the extended Selberg class. De Bruijn's upper bound of Λ ≤ 1 / 2 {\displaystyle
De_Bruijn–Newman_constant
Spanish physicist (born 1950)
Barcelona. His well-regarded book Ten Physical Applications of Spectral Zeta Functions (1995) was written during the period of his association with CSIC. Although
Emilio_Elizalde
Mathematics award
the original on 8 April 2022. Retrieved 7 April 2019. "Remembering Atle Selberg, 1917–2007" (PDF). Ams.org. Archived (PDF) from the original on 23 November
Fields_Medal
American-Israeli mathematician
automorphic L-functions in vertical strips. J. Amer. Math. Soc. 14 (2001) 79–107. MR 1800349 with Stephen D. Miller: Riemann's zeta function and beyond.
Stephen_Gelbart
Proof that only uses basic techniques
roughly equivalent to a theorem about an analytic function, the theorem that Riemann's zeta function has no roots on a certain line. A proof of such a
Elementary_proof
German mathematician (1911–1992)
also concerned with automorphic functions in several variables, Siegel modular functions, and associated zeta functions. Maass, Hans (1949), "Über eine
Hans_Maass
Type of mathematical group
{\displaystyle D_{F}} its discriminant, and ζ F {\displaystyle \zeta _{F}} its Dedekind zeta function. Let Γ O {\displaystyle \Gamma _{\mathcal {O}}} be the arithmetic
Arithmetic_Fuchsian_group
American mathematician (born 1942)
finite groups, special functions, algebraic graph theory, zeta functions of graphs, arithmetical quantum chaos, and the Selberg trace formula. Terras was
Audrey_Terras
Theorem in abstract algebra
strategy for proving local and global Langlands conjectures using the Arthur–Selberg trace formula, but in order for this approach to work, the geometric sides
Fundamental lemma (Langlands program)
Fundamental_lemma_(Langlands_program)
American mathematician (born 1947)
Lett. 6 (1999), no. 3–4, Anshel, Michael; Goldfeld, Dorian Zeta functions, one-way functions, and pseudorandom number generators. Duke Math. J. 88 (1997)
Dorian_M._Goldfeld
Series whose partial sums eventually only have a fixed number of terms after cancellation
CreateSpace, 2008, page 85 Weil, André (1989). "Prehistory of the zeta-function". In Aubert, Karl Egil; Bombieri, Enrico; Goldfeld, Dorian (eds.). Number
Telescoping_series
Canadian mathematician
which the Hasse–Weil zeta functions of arithmetic quotients of the upper half plane are identified with L {\displaystyle L} -functions occurring in Hecke's
Robert_Langlands
French mathematician (born 1962)
with complex multiplication, a far-reaching generalization of the Chowla-Selberg formula. A proof of Perrin-Riou's conjectural explicit reciprocity law
Pierre_Colmez
Mathematical conjecture about elliptic curves
in front. It is mentioned in J. Tate, Algebraic cycles and poles of zeta functions in the volume (O. F. G. Schilling, editor), Arithmetical Algebraic Geometry
Sato–Tate_conjecture
number theory". In Balasubramanian, B.; Srinivas, K. (eds.). The Riemann zeta function and related themes: papers in honour of Professor K. Ramachandra. Ramanujan
Four_exponentials_conjecture
Mathematical function
they lead to harmonic Maass forms. The evaluation of the Weierstrass zeta function at the Eichler integral of the weight 2 new form corresponding to a
Harmonic_Maass_form
Mathematical seminars held in Paris since 1948
fonctions algébriques de caractéristique p, I, d'après Weil (local zeta-function) Roger Godement, Groupe complexe unimodulaire, I : Les représentations
Séminaire_Nicolas_Bourbaki
On eigenvalues of random matrices
= π N ∏ k = 1 N k ! {\displaystyle Z=\pi ^{N}\prod _{k=1}^{N}k!} is a Selberg integral. Ignoring the term Z {\displaystyle Z} , the rest of the formula
Circular_law
British theoretical physicist and mathematician (1923–2020)
distribution of the zeros of the Riemann zeta function. He showed his formula to the mathematician Atle Selberg, who said that it looked like something
Freeman_Dyson
Israeli mathematician
He has contributed to one of the discoveries concerning the Riemann zeta function, namely, that the Riemann zeros appear to display the same statistics
Zeev_Rudnick
Discrete analog of a derivative
Blagouchine (2018). "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF). Integers (Electronic Journal of Combinatorial Number Theory)
Finite_difference
American mathematician
point-counting records, and average polynomial-time algorithms for computing zeta functions of hyperelliptic curves over finite fields, developed jointly with David
Andrew Sutherland (mathematician)
Andrew_Sutherland_(mathematician)
Ilya Piatetski-Shapiro and Shalika pertain to L-functions of pairs, called the Rankin-Selberg L-functions, attached to representations of GL ( n ) {\displaystyle
Hervé_Jacquet
Matrix-valued random variable
number theory, the distribution of zeros of the Riemann zeta function (and other L-functions) is modeled by the distribution of eigenvalues of certain
Random_matrix
Hungarian American mathematician (1924–2016)
number theory: With special emphasis on the theory of the Zeta functions of number fields and function fields. Jones Letter Service. ASIN B0007FPWF6. (453 pp
Steven_Gaal
non-additif (derived functors) Roger Godement, Les fonctions zêta des algèbres simples, I (zeta-function of a simple algebra) Michel A. Kervaire, L'homotopie
Séminaire Nicolas Bourbaki (1950–1959)
Séminaire_Nicolas_Bourbaki_(1950–1959)
Japanese mathematician and professor
mathematical physics, and number theory, particularly the theory of zeta functions. He stated the study of representation theory and invariant theory of
Masato_Wakayama
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SELBERG ZETA-FUNCTION
SELBERG ZETA-FUNCTION
SELBERG ZETA-FUNCTION
SELBERG ZETA-FUNCTION
SELBERG ZETA-FUNCTION
SELBERG ZETA-FUNCTION
SELBERG ZETA-FUNCTION
SELBERG ZETA-FUNCTION
SELBERG ZETA-FUNCTION
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