Riemann zeta function

Riemann zeta function
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Name	Riemann zeta function
Field	Complex analysis, Analytic number theory
Introduced	1859
Introduced by	Bernhard Riemann

Riemann zeta function is a complex function originally defined for complex numbers with real part greater than one by an infinite series and extended to a meromorphic function on the complex plane, pivotal in Bernhard Riemann's 1859 memoir. It connects results of Leonhard Euler on infinite series, the distribution of Prime numbers explored by Adrien-Marie Legendre and Carl Friedrich Gauss, and conjectures central to modern mathematics such as the Riemann Hypothesis posited by Bernhard Riemann. The function plays a foundational role in analytic number theory, influencing work of G. H. Hardy, John von Neumann, Alan Turing, and institutions like the Princeton University mathematics department and the Clay Mathematics Institute.

Definition and basic properties

For complex s with real part greater than one the function is given by the absolutely convergent Dirichlet series ζ(s) = Σ_{n=1}^∞ n^{-s}, reflecting methods used by Leonhard Euler in the study of the Basel problem and the harmonic series studied by Nicole Oresme. Its Euler product over primes, ζ(s) = Π_{p prime} (1 − p^{-s})^{-1}, links the function to prime distribution investigated by Peter Gustav Lejeune Dirichlet and later formalized in proofs by Jacques Hadamard and Charles de la Vallée Poussin. The series and product reveal multiplicative properties leveraged in proofs by Paul Erdős and Atle Selberg and underpin explicit formulas developed by Harald Bohr and George Pólya.

Analytic continuation and functional equation

Bernhard Riemann constructed an analytic continuation of the function to a meromorphic function on the complex plane with a simple pole at s = 1, building on techniques from Augustin-Louis Cauchy and Karl Weierstrass. The functional equation relates values at s and 1−s via the completed zeta function Λ(s) = π^{-s/2} Γ(s/2) ζ(s), invoking the Gamma function introduced by Adrien-Marie Legendre and studied by Niels Henrik Abel and Sofia Kovalevskaya. This symmetry under s ↦ 1−s informed investigations by Bernhard Riemann and later refinements by Edmond Maillet and Ernst Eduard Kummer in the context of modular transformations examined by Henri Poincaré and Felix Klein.

Zeros and the Riemann Hypothesis

Nontrivial zeros of the function lie in the critical strip 0 < Re(s) < 1 and are conjectured by Bernhard Riemann to have real part 1/2, a statement known as the Riemann Hypothesis, one of the Millennium Prize Problems highlighted by the Clay Mathematics Institute. The distribution of zeros connects to Spectral theory used by Hilbert and Pólya in their conjectural approaches and to computational verifications by Alan Turing, Andrew Odlyzko, and teams at Bell Labs and University of Minnesota. Results by G. H. Hardy, Atle Selberg and Enrico Bombieri established infinitely many zeros on the critical line and conditional bounds used in sieving techniques by Daniel Goldston and Jerzy Kaczorowski. Relationships between zeros and random matrix ensembles were proposed by Freeman Dyson and developed by Michael Berry and Jon Keating.

Special values and relationships to number theory

Values of the function at integers connect to Bernoulli numbers studied by Jakob Bernoulli and to evaluations like ζ(2) = π^2/6 found by Leonhard Euler, further related to polylogarithms examined by Jonas Persson and Leonard Lewin. Rational values at negative integers correspond to arithmetic invariants exploited by Kummer and Emil Artin in class field theory, and special values at even positive integers relate to periods appearing in the work of Alexander Grothendieck and Pierre Deligne. Connections to divisor problems studied by Dirichlet and Srinivasa Ramanujan appear in explicit formulae used by I. M. Vinogradov and Atle Selberg in analytic estimates.

Applications and connections (prime number theorem, L-functions)

The zeta function underlies the proof of the Prime number theorem established independently by Jacques Hadamard and Charles de la Vallée Poussin via nonvanishing of ζ(s) on Re(s)=1, and it appears in explicit formulas connecting primes to zeros developed by Riemann and refined by John von Neumann and Harold Davenport. Generalizations to automorphic contexts informed the Langlands program proposed by Robert Langlands and influenced studies by Andrew Wiles and Richard Taylor on modularity and elliptic curves. In physics, analogies with spectral traces in Albert Einstein's and Paul Dirac's frameworks and statistical models used by Eugene Wigner and Freeman Dyson link zeta phenomena to quantum chaos explored by Michael Berry.

Generalizations and extensions (Dirichlet, Dedekind, Hurwitz)

Dirichlet L-functions introduced by Peter Gustav Lejeune Dirichlet extend multiplicative ideas to characters and yield Dirichlet’s theorem on primes in arithmetic progressions examined by Adrien-Marie Legendre and Srinivasa Ramanujan. Dedekind zeta functions associated to number fields were developed by Richard Dedekind to study class numbers, later used by Hecke and Emil Artin in reciprocity laws and class field theory. The Hurwitz zeta function generalizes the argument by a shift parameter and has been applied in work by Hurwitz himself and by analysts such as L. D. Landau and Tom M. Apostol. Further extensions include Selberg zeta functions studied by Atle Selberg, motivic L-functions in conjectures of Pierre Deligne and Alexander Grothendieck, and adelic formulations used by Andre Weil and researchers in the Langlands program.

Category:Complex analysis Category:Analytic number theory Category:Bernhard Riemann