Analytic number theory
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| Analytic number theory | |
|---|---|
| Name | Analytic number theory |
| Discipline | Mathematics |
| Subdiscipline | Number theory |
Analytic number theory is a branch of mathematics using methods from complex analysis, harmonic analysis, and probability to study integers, arithmetic functions, and prime distribution. It draws on tools developed in the works of Bernhard Riemann, G. H. Hardy, John Edensor Littlewood, and Atle Selberg, and has influenced research at institutions such as University of Göttingen, Trinity College, Cambridge, and Institute for Advanced Study. Modern developments connect research programs at Princeton University, University of Cambridge, École Normale Supérieure, and University of Oxford with conjectures named after Riemann Hypothesis, Goldbach conjecture, and Twin prime conjecture.
History and development
The subject traces origins to results by Leonhard Euler on Bernoulli numbers and the Euler product alongside Carl Friedrich Gauss's work that influenced later studies such as the Prime Number Theorem proven with methods of Jacques Hadamard and Charles-Jean de la Vallée Poussin, and expanded by collaborations between Hardy and Littlewood and later refinements by Selberg and Atle Selberg's elementary proof efforts. Nineteenth-century advances by Riemann introduced the Riemann zeta function, which inspired twentieth-century programs at University of Chicago and Massachusetts Institute of Technology where analysts like Ernst Kummer and John von Neumann influenced computational and theoretical techniques. Twentieth-century breakthroughs involved work by Ivan Vinogradov, Heath-Brown, and Paul Erdős while late-century progress connected the field to research at Clay Mathematics Institute and prizes such as the Fields Medal.
Fundamental tools and techniques
Analysts routinely employ the Riemann zeta function, Dirichlet L-series, Fourier transform, Mellin transform, and contour integration techniques developed in the traditions of Augustin-Louis Cauchy, Bernhard Riemann, Gaston Darboux, and Émile Borel, with functional equations studied in the spirit of André Weil and Harish-Chandra. Tauberian theorems derived from work by G. H. Hardy and J. E. Littlewood interact with techniques from John von Neumann style spectral theory and Atle Selberg's trace formula, while explicit formulae link zeros of zeta-type functions to prime counting following traditions at University of Göttingen and University of Cambridge. Probabilistic methods inspired by Paul Erdős and Mark Kac combine with exponential sum bounds pioneered by I. M. Vinogradov and Nikolai Korobov, and large sieve inequalities associated with Enrico Bombieri and Yu. V. Linnik are standard.
Prime number theorems and distribution of primes
Work on prime distribution follows the roadmap from Riemann's 1859 memoir through the formal proofs of the Prime Number Theorem by Hadamard and de la Vallée Poussin and later refinements by Atle Selberg and Chen Jingrun's contributions to prime gaps. Modern results on primes in arithmetic progressions build on Dirichlet's theorem with analytic refinements by G. H. Hardy, John Littlewood, Enrico Bombieri, and the Bombieri–Vinogradov theorem connected to research at Institute for Advanced Study. The study of small gaps and bounded gaps owes advances to collaborative projects at Zhejiang University and Princeton University culminating in work by Yitang Zhang and later collaborative efforts involving Terence Tao and James Maynard.
L-functions and modular forms
L-functions such as Dirichlet L-series and the Riemann zeta function relate to automorphic forms studied in the framework developed by Atle Selberg and Robert P. Langlands and pursued at Institute for Advanced Study and Harvard University. The theory of modular forms advanced by Jean-Pierre Serre, Pierre Deligne, and Andrew Wiles connects to conjectures like the Taniyama–Shimura–Weil conjecture and results such as the proof of Fermat's Last Theorem at Princeton University and University of Cambridge. Higher-degree L-functions and functoriality are central to programs proposed by Langlands and further developed by researchers at Institut des Hautes Études Scientifiques and Max Planck Institute.
Additive number theory and sieve methods
Additive problems like the Goldbach conjecture and representations of integers draw on methods by Ivan Vinogradov, Harald Helfgott, and John Green collaborating with Terence Tao in additive combinatorics, with key inputs from research at University of Cambridge, University of Oxford, and University of California, Los Angeles. Sieve theory owes its foundations to Brun and further refinements by Atle Selberg, Heini Halberstam, H.-E. Richert, and contemporary developments by Daniel Goldston and János Pintz; techniques such as the Large Sieve and Selberg sieve permeate work at University of Chicago and Stanford University.
Important problems and conjectures
Major open problems include the Riemann Hypothesis, the Generalized Riemann Hypothesis, the Goldbach conjecture, the Twin prime conjecture, problems about gaps advanced by Yitang Zhang, and the distributional conjectures studied by Graham, Kolesnik and Montgomery and others at University of Michigan and Rutgers University. Conjectures relating L-functions and automorphic forms appear in the Langlands program articulated by Robert Langlands and pursued at centers including Princeton University and Institut des Hautes Études Scientifiques; further targets include subconvexity estimates pursued by P. Sarnak and equidistribution results connected to Peter Sarnak's collaborations at Harvard University and Princeton University.