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| Extended Riemann Hypothesis | |
|---|---|
| Name | Extended Riemann Hypothesis |
| Field | Number theory |
| Conjectured by | Bernhard Riemann |
| Date | 1859 |
| Status | Open problem |
Extended Riemann Hypothesis The Extended Riemann Hypothesis is a central conjecture in analytic number theory concerning the zeros of Dirichlet L-functions and their generalizations; its resolution would impact results in prime distribution, algebraic number theory, and cryptography. It generalizes statements from Bernhard Riemann and interacts with work by G. H. Hardy, John Edensor Littlewood, Atle Selberg, Alan Turing, and Enrico Bombieri. The conjecture appears across literature associated with institutions such as Princeton University, University of Cambridge, Institute for Advanced Study, ETH Zurich, and journals linked to American Mathematical Society.
The formal statement asserts that for every primitive Dirichlet character modulo q studied by Carl Friedrich Gauss and later formalized by Dirichlet and Leopold Kronecker, every nontrivial zero of the associated Dirichlet L-function L(s, χ) has real part 1/2, mirroring the claim first proposed by Bernhard Riemann for the Riemann zeta function. Variants extend the assertion to Hecke L-functions associated to number fields investigated by Erich Hecke and to automorphic L-functions arising in the Langlands program developed by Robert Langlands; such refinements connect to conjectures advanced at gatherings like conferences at Institute for Advanced Study and results pursued at IHÉS. Precise formulations draw on analytic techniques connected to Atle Selberg's trace formula and to explicit formulae influenced by John von Neumann's spectral perspectives.
Origins trace to Bernhard Riemann's 1859 memoir, later expanded by Peter Gustav Lejeune Dirichlet's 19th-century work on characters and by Leopold Kronecker's algebraic viewpoints; the Dirichlet L-series framework grew through contributions by Erich Hecke and by Hendrik Lorentz-era analysts like G. H. Hardy and John Edensor Littlewood. In the 20th century, computational efforts by Alan Turing, Andrew Odlyzko and numerical projects at University of Cambridge and Princeton University tested zeros for ζ(s) and for specific Dirichlet L-series, while theoretical advances by Atle Selberg, Harold Davenport, Heath-Brown, and Enrico Bombieri produced conditional results assuming the hypothesis. The expansion into automorphic settings owes much to Robert Langlands and to work at institutions like Institute for Advanced Study and ETH Zurich, influencing collaborative programs with researchers affiliated with Harvard University and Massachusetts Institute of Technology.
The Extended Riemann Hypothesis implies strong versions of the Chebotarev density theorem's effective bounds as applied in contexts studied by Emmy Noether and Richard Dedekind, refines error terms in the Prime Number Theorem variants examined by Bernhard Riemann and Jacques Hadamard, and yields consequences for distributional results reminiscent of conjectures by Paul Erdős and Atle Selberg. It is intimately related to the Generalized Riemann Hypothesis and to the automorphic predictions in the Langlands program proposed by Robert Langlands, and it influences cryptographic hardness assumptions involved in protocols analyzed by investigators at National Institute of Standards and Technology and by researchers affiliated with RSA Security. Equivalences and implications connect to conjectures proposed by Peter Sarnak and to spectral questions studied by Yakov Sinai and Barry Mazur.
Extensive computational verification for many Dirichlet characters has been carried out by teams led by Alan Turing, Andrew Odlyzko, H. M. Edwards, and groups at Princeton University and University of Cambridge, checking zeros up to high heights for principal and nonprincipal characters and for specific moduli such as those studied in datasets produced at IBM Research and by researchers at Los Alamos National Laboratory. Numerical methods derive from work by Atle Selberg's evaluations, John von Neumann's computational frameworks, and algorithmic improvements influenced by Srinivasa Ramanujan-inspired identities and by software projects at Mathematical Reviews and Wolfram Research. Empirical support parallels results for ζ(s) where zeros up to very large heights satisfy the 1/2 line, though counterexamples remain elusive and rigorous proofs remain outstanding despite datasets maintained by National Institute of Standards and Technology and collaborative teams at ETH Zurich.
Many conditional theorems assume the Extended Riemann Hypothesis to prove effective bounds in algebraic number theory and analytic estimates, paralleling unconditional theorems by Heath-Brown, Harold Davenport, and Atle Selberg. Zero-free regions for Dirichlet L-functions away from the critical line were established in work connected to Charles de la Vallée Poussin and refined by Enrico Bombieri and D. A. Burgess; explicit zero-density results derive from techniques pioneered by I. M. Vinogradov and Arkady Selberg. Results on low-lying zeros and on low-level density connect to random matrix models advanced by Freeman Dyson and Hugh Montgomery, with theoretical bridges developed by P. X. Gallagher and Peter Sarnak.
Generalizations include the Generalized Riemann Hypothesis for L-functions of number fields studied by Richard Dedekind and Emil Artin, the hypothesis for Hecke L-functions tied to Erich Hecke and to class field theory of David Hilbert, and automorphic extensions framed in the Langlands program by Robert Langlands and pursued by researchers at Princeton University and Institute for Advanced Study. Variants involve the Grand Riemann Hypothesis addressing all automorphic L-functions considered in contexts with contributions from Pierre Deligne, Andrew Wiles, and Gerald B. Folland, and statistical generalizations inspired by random matrix theory by Freeman Dyson, P. J. Forrester, and Nicholas Katz.