In mathematics, a Beurling zeta function is an analogue of the Riemann zeta function where the ordinary primes are replaced by a set of Beurling generalized primes: any sequence of real numbers greater than 1 that tend to infinity. These were introduced by Beurling (1937).
A Beurling generalized integer is a number that can be written as a product of Beurling generalized primes[definition needed]. Beurling generalized the usual prime number theorem to Beurling generalized primes. He showed that if the number N(x) of Beurling generalized integers less than x is of the form N(x) = Ax + O(x log−γx) with γ > 3/2 then the number of Beurling generalized primes less than x is asymptotic to x/log x, just as for ordinary primes, but if γ = 3/2 then this conclusion need not hold.
Bateman, Paul T.; Diamond, Harold G. (1969), "Asymptotic distribution of Beurling's generalized prime numbers", in LeVeque, William Judson (ed.), Studies in Number Theory, M.A.A. studies in mathematics, vol. 6, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), pp. 152–210, ISBN978-0-13-541359-3, MR 0242778
Beurling, Arne (1937), "Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. I", Acta Mathematica (in French), Springer Netherlands, 68: 255–291, doi:10.1007/BF02546666, ISSN 0001-5962, Zbl 0017.29604
beurling, zeta, function, mathematics, analogue, riemann, zeta, function, where, ordinary, primes, replaced, beurling, generalized, primes, sequence, real, numbers, greater, than, that, tend, infinity, these, were, introduced, beurling, 1937, beurling, general. In mathematics a Beurling zeta function is an analogue of the Riemann zeta function where the ordinary primes are replaced by a set of Beurling generalized primes any sequence of real numbers greater than 1 that tend to infinity These were introduced by Beurling 1937 A Beurling generalized integer is a number that can be written as a product of Beurling generalized primes definition needed Beurling generalized the usual prime number theorem to Beurling generalized primes He showed that if the number N x of Beurling generalized integers less than x is of the form N x Ax O x log gx with g gt 3 2 then the number of Beurling generalized primes less than x is asymptotic to x log x just as for ordinary primes but if g 3 2 then this conclusion need not hold See also EditAbstract analytic number theoryReferences EditBateman Paul T Diamond Harold G 1969 Asymptotic distribution of Beurling s generalized prime numbers in LeVeque William Judson ed Studies in Number Theory M A A studies in mathematics vol 6 Math Assoc Amer distributed by Prentice Hall Englewood Cliffs N J pp 152 210 ISBN 978 0 13 541359 3 MR 0242778 Beurling Arne 1937 Analyse de la loi asymptotique de la distribution des nombres premiers generalises I Acta Mathematica in French Springer Netherlands 68 255 291 doi 10 1007 BF02546666 ISSN 0001 5962 Zbl 0017 29604 This mathematical analysis related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Beurling zeta function amp oldid 1013648079, wikipedia, wiki, book, books, library,
