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| Erdős–Kac | |
|---|---|
| Name | Erdős–Kac theorem |
| Field | Number theory |
| Statement | Distribution of number of distinct prime factors |
| Introduced | 1939 |
| Authors | Paul Erdős, Mark Kac |
Erdős–Kac.
The Erdős–Kac theorem is a central result in Number theory and Probability theory asserting that the normal distribution governs the number of distinct prime factors of a typical integer; it connects ideas from Paul Erdős, Mark Kac, Gábor Szegő, Wacław Sierpiński, Atle Selberg and Srinivasa Ramanujan to later developments by Andrew Granville, K. Ramachandra, Kurt Mahler, Heini Halberstam and Hans Rademacher. The theorem sits at the intersection of research by Émile Borel, Andrey Kolmogorov, Norbert Wiener, Alfréd Rényi, Paul Turán and Harald Cramér and influenced work by John Littlewood, Alan Baker, Enrico Bombieri, Ralph K. Guy and Serge Lang.
The result was formulated by Paul Erdős and Mark Kac in the late 1930s and popularized in a sequence of papers and lectures alongside contemporaneous contributions from György Polya and George Pólya; it was proved using ideas that trace back to Srinivasa Ramanujan and the heuristic statistics of G.H. Hardy. The precise statement: for large n the random variable ω(n), the number of distinct prime factors of n, after centering by log log n and scaling by sqrt(log log n), converges in distribution to the standard normal distribution introduced by Carl Friedrich Gauss and formalized by Andrey Kolmogorov and Paul Lévy. The formal limit theorem ties to classical central limit theorems of Aleksandr Lyapunov and Miklós Csörgő and was influenced by limit conceptions from William Feller and Sergei Bernstein.
The theorem inaugurated modern Probabilistic number theory by applying probabilistic models from Alfréd Rényi, Andrey Kolmogorov, William Feller and Andrei Markov to multiplicative functions studied by Srinivasa Ramanujan, G.H. Hardy, John Littlewood and Edmund Landau. It uses sieve ideas developed by V. Balakrishnan, Bruno de Finetti-style heuristics and analytic inputs from Atle Selberg and Rosser and Schoenfeld bounds on primes, linking to prime distribution work of Bernhard Riemann, G. H. Hardy, Jacques Hadamard, Charles de la Vallée Poussin, Harold L. Montgomery and Andrew Granville. The approach situates the theorem within frameworks advanced by Paul Turán, Alain Selberg, Enrico Bombieri and Dorian Goldfeld.
Proofs combine characteristic-function techniques of Paul Lévy and Andrey Kolmogorov, truncation and independence approximations inspired by Alfréd Rényi and sieve-theoretic inputs from Atle Selberg and Brun. Key steps use approximation of ω(n) by sums of indicator variables for divisibility by primes p ≤ z, employing independence heuristics akin to methods of Erdős–Kac and refinements by Andrew Granville, Heini Halberstam and H. Maier. Analytic estimates rely on prime counting functions studied by Bernhard Riemann, Adrien-Marie Legendre, Gauss and explicit bounds by Rosser and Schoenfeld, together with moment methods from Pál Turán and the method of moments popularized by Wassily Hoeffding and Sergei Bernstein.
Generalizations extend to additive and multiplicative functions in the spirit of Elliott, Halberstam, Granville and Soundararajan, K. Ramachandra and E. Wirsing, including versions for Ω(n) (counting prime factors with multiplicity) and for values of arithmetical functions studied by Paul Erdős, Heini Halberstam, Andrew Granville and K. Soundararajan. Refinements obtain rates of convergence and large deviations via work by Eugene Kac, Pál Turán, H. Maier and Elliott, while connections to distributional results of Gál and limit theorems developed by William Feller and Andrey Kolmogorov produced local limit theorems and moderate deviation estimates. Multidimensional and conditional extensions relate to conjectures by Graham, Knuth and Patashnik and conditional analyses assuming the Riemann hypothesis studied by Atle Selberg and Enrico Bombieri.
Applications appear across analytic investigations by John Littlewood, G.H. Hardy, Paul Erdős, Andrew Granville and K. Ramachandra in problems on normal orders, distribution of smooth numbers, and value distribution of multiplicative functions explored by D. H. Lehmer, G. H. Hardy, J. E. Littlewood, József Turán and Paul Turán. Related results include the Prime Number Theorem (proven by Jacques Hadamard and Charles de la Vallée Poussin), the Erdős–Kac-type law for polynomials over finite fields studied by Emil Artin and André Weil, and probabilistic models developed by Ben Green, Terence Tao and Timothy Gowers in additive combinatorics. The theorem informs heuristics in computational tasks addressed by Alan Turing, Donald Knuth, Richard Brent and applied cryptography research influenced by Ronald Rivest, Adi Shamir and Leonard Adleman.
Empirical verification draws on tabulations of ω(n) and Ω(n) implemented in algorithms by Donald Knuth, Richard Brent, D. J. Bernstein and computational projects associated with The On-Line Encyclopedia of Integer Sequences and databases curated by Neil Sloane; large-scale tests used prime tables stemming from works by D. H. Lehmer and bounds by Rosser and Schoenfeld. Numerical studies by Paul Erdős, Mark Kac, Andrew Granville and K. Soundararajan confirm convergence rates and deviations consistent with classical limit theorems of Paul Lévy and William Feller and with heuristic models promoted by Alfréd Rényi and Andrey Kolmogorov.
Category:Theorems in number theory