Zeta function
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| Zeta function | |
|---|---|
| Name | Zeta function |
| Field | Mathematics |
| Introduced | 1859 |
| Notable | Bernhard Riemann, Leonhard Euler, George Pólya |
Zeta function
The zeta function is a complex-valued function originally defined by a Dirichlet series that encodes deep arithmetic, analytic, and spectral information; it connects figures such as Leonhard Euler, Bernhard Riemann, G. H. Hardy, John von Neumann, and Andrey Kolmogorov with problems in prime distribution, Random matrix theory, Quantum chaos, and Statistical mechanics. Its study intersects work by Srinivasa Ramanujan, Atle Selberg, Harold Davenport, Enrico Bombieri, and institutions such as Institute for Advanced Study, Princeton University, University of Göttingen, and École Normale Supérieure.
Definition and Basic Properties
The classical prototype is the Dirichlet series ζ(s)=∑_{n=1}^∞ n^{-s} introduced by Leonhard Euler and systematized by Bernhard Riemann; it converges for complex s with Re(s)>1 and exhibits Euler product factorization over primes via ∏_{p}(1−p^{-s})^{-1}, connecting work by Euclid, Adrien-Marie Legendre, Karl Gauss, and Dirichlet to multiplicative number theory. Fundamental properties include absolute convergence in the half-plane Re(s)>1, meromorphic continuation with a simple pole, and relationships with special values at integers studied by Leonhard Euler, Kummer, Leopold Kronecker, and Emil Artin. Functional relations and symmetry constraints tie this prototype to spectral results proved by Atle Selberg and to explicit formulae used by G. H. Hardy and J. E. Littlewood in analytic estimates.
Examples and Special Cases
Classical examples and variants include the Riemann zeta studied by Bernhard Riemann, the Hurwitz zeta associated to Adolf Hurwitz, Dirichlet L-series attached to characters introduced by Peter Gustav Lejeune Dirichlet, and Dedekind zeta functions of number fields developed by Richard Dedekind. Epstein zeta functions arising in quadratic forms link to work by Paul Epstein and to theta series of Carl Gustav Jacob Jacobi and Srinivasa Ramanujan. Hasse–Weil zeta functions for algebraic varieties connect to conjectures of André Weil, proven cases by Pierre Deligne, and to Alexander Grothendieck's cohomological framework. Special values at nonpositive integers relate to Bernoulli numbers studied by Jacob Bernoulli and to K-theory investigations by Daniel Quillen and John Milnor.
Analytic Continuation and Functional Equation
Riemann established analytic continuation and a functional equation for the prototype using techniques later linked to Fourier analysis by Joseph Fourier and to modular forms studied by Srinivasa Ramanujan and Erich Hecke. The functional equation relates values at s and 1−s and involves the Gamma function analyzed by Adrien-Marie Legendre and extended by Émile Picard; proof techniques employ Mellin transforms used by Issai Schur and contour integration methods refined by Hermann Weyl. For Dedekind and Hasse–Weil zeta functions, functional equations reflect duality principles in Alexander Grothendieck's étale cohomology and reciprocity laws developed by Emil Artin and Jean-Pierre Serre.
Zeros and the Riemann Hypothesis
Zero distribution of the classical series is central to questions proposed by Bernhard Riemann and later popularized by David Hilbert and Atle Selberg; nontrivial zeros lie in the critical strip 0
Applications in Number Theory and Physics
Zeta and L-functions underpin proofs of prime distribution results by Jacques Hadamard and Charles de la Vallée Poussin and enter reciprocity phenomena explored by Ernst Kummer and Heinrich Weber. In physics, spectral zeta regularization appears in quantum field theory applications by Stephen Hawking, Casimir effect computations credited to Hendrik Casimir, and partition-function approaches in Ludwig Boltzmann-inspired statistical mechanics. Connections to quantum chaos relate to work by Michael Berry and Mark Kelbert and to semiclassical analysis developed by Martin Gutzwiller and Mikhail Shubin.
Generalizations and Related Zeta Functions
Generalizations include Selberg zeta functions for manifolds introduced by Atle Selberg, Ihara zeta functions for graphs initiated by Yasutaka Ihara, and p-adic zeta functions developed by Kenkichi Iwasawa and Serre; automorphic L-functions studied by Robert Langlands are central to the Langlands program promoted by Andrew Wiles, Pierre Deligne, and James Arthur. Further relatives are motivic L-functions in the work of Alexander Grothendieck and Pierre Deligne, topological zeta functions appearing in singularity theory of Bernard Teissier, and L-series in arithmetic geometry advanced by Gerd Faltings and Alexander Beilinson.