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Properties of the Beurling generalized primes
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2011 (English)In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 131, no 1, 45-58 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we prove a generalization of Mertens' theorem to Beurling primes, namely that lim(x ->infinity) 1/Inx Pi(p <= x)(1 - p(-1))(-1) = Ae(gamma), where gamma is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit M = lim(x ->infinity) (Sigma(p <= x) p(-1) - In(Inx)) exists. We also show that this limit coincides with lim(alpha -> 0+)(Sigma(p) p(-1)(In p)(-alpha) - 1/alpha); for ordinary primes this claim is called Meissel's theorem. Finally, we will discuss a problem posed by Beurling, namely how small vertical bar N(x)-[4]vertical bar can be made for a Beurling prime number system Q not equal P. where P is the rational primes. We prove that for each c > 0 there exists a Q such that vertical bar N(x) - [x]vertical bar < cInx and conjecture that this is the best possible bound.

Place, publisher, year, edition, pages
2011. Vol. 131, no 1, 45-58 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-24337DOI: 10.1016/j.jnt.2010.06.014ISI: 000283401800004OAI: oai:DiVA.org:kth-24337DiVA: diva2:346650
Funder
Knut and Alice Wallenberg Foundation
Note
Updated from submitted to published. QC 20120327Available from: 2010-09-02 Created: 2010-09-02 Last updated: 2017-12-12Bibliographically approved
In thesis
1. Problems in Number Theory related to Mathematical Physics
Open this publication in new window or tab >>Problems in Number Theory related to Mathematical Physics
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of an introduction and four papers. All four papers are devoted to problems in Number Theory. In Paper I, a special class of local ζ-functions is studied. The main theorem states that the functions have all zeros on the line Re(s)=1/2.This is a natural generalization of the result of Bump and Ng stating that the zeros of the Mellin transform of Hermite functions have Re(s)=1/2.In Paper II and Paper III we study eigenfunctions of desymmetrized quantized cat maps.If N denotes the inverse of Planck's constant, we show that the behavior of the eigenfunctions is very dependent on the arithmetic properties of N. If N is a square, then there are normalized eigenfunctions with supremum norm equal to , but if N is a prime, the supremum norm of all eigenfunctions is uniformly bounded. We prove the sharp estimate for all normalized eigenfunctions and all $N$ outside of a small exceptional set. For normalized eigenfunctions of the cat map (not necessarily desymmetrized), we also prove an entropy estimate and show that our functions satisfy equality in this estimate.We call a special class of eigenfunctions newforms and for most of these we are able to calculate their supremum norm explicitly.For a given , with k>1, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about and the supremum norm of the newforms in the second half have at most three different values, all of the order . The only dependence of A is that the normalization factor is different if A has eigenvectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms.In Paper IV we prove a generalization of Mertens' theorem to Beurling primes, namely that

\lim_{n \to \infty}\frac{1}{\ln n}\prod_{p \leq n} \left(1-p^{-1}\right)^{-1}=Ae^{\gamma}where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit exists. We also show that this limit coincides with ; for ordinary primes this claim is called Meissel's theorem. Finally we will discuss a problem posed by Beurling, namely how small |N(x)-[x] | can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c>0 there exists a Q such that |N(x)-[x] |

Place, publisher, year, edition, pages
Stockholm: KTH, 2008. viii, 28 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 08:12
Keyword
number theory, mathetical physics, local zeta functions, cat maps, Beurling primes
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-9514 (URN)978-91-7415-177-0 (ISBN)
Public defence
2008-12-08, F3, KTH, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20100902Available from: 2008-11-21 Created: 2008-11-11 Last updated: 2010-09-02Bibliographically approved

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