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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.
Mathematics
##### Identifiers
ISI: 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 $N^{1/4}$, but if N is a prime, the supremum norm of all eigenfunctions is uniformly bounded. We prove the sharp estimate $\|\psi\|_\infty=O(N^{1/4})$ 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 $N=p^k$, 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 $2/\sqrt{1\pm 1/p}$ and the supremum norm of the newforms in the second half have at most three different values, all of the order $N^{1/6}$. 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}$\lim_{n \to \infty}\frac{1}{\ln n}\prod_{p \leq n} \left(1-p^{-1}\right)^{-1}=Ae^{\gamma},$where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit $M=\lim_{n\to\infty}\left(\sum_{p\le n}p^{-1}-\ln(\ln n)\right)$exists. We also show that this limit coincides with $\lim_{\alpha\to 0^+} \left(\sum_p p^{-1}(\ln p)^{-\alpha}-1/\alpha\right)$ ; 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
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)
##### Note
QC 20100902Available from: 2008-11-21 Created: 2008-11-11 Last updated: 2010-09-02Bibliographically approved

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