Hostname: page-component-669899f699-ggqkh Total loading time: 0 Render date: 2025-04-25T19:28:55.257Z Has data issue: false hasContentIssue false
  • English
  • Français

On Mertens' Theorem for Beurling Primes

Published online by Cambridge University Press:  20 November 2018

Paul Pollack
Paul Pollack*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2 e-mail: pollack@math.ubc.ca
Rights & Permissions [Opens in a new window]

Abstract.

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $1\,<\,{{p}_{1}}\,\le \,{{p}_{2}}\,\le \,{{p}_{3}}\,\le \,...$ be an infinite sequence $P$ of real numbers for which ${{p}_{i}}\,\to \,\infty $, and associate with this sequence the Beurling zeta function$\zeta P\left( s \right)\,:=\,{{\prod\nolimits_{i=1}^{\infty }{\left( 1\,-\,p_{i}^{-s} \right)}}^{-1}}$. Suppose that for some constant $A\,>\,0$, we have $\zeta P\left( s \right)\tilde{\ }A/\left( s-1 \right),\ \text{as}\,s\,\downarrow \,1$. We prove that $P$ satisfies an analogue of a classical theorem of Mertens: ${{\prod{_{{{p}_{i}}\le x}\left( 1\,-\,{1}/{{{p}_{i}}}\; \right)}}^{-1}}\,\sim \,A{{\text{e}}^{\gamma }}\,\log \,x$, as $x\,\to \,\infty $. Here $\text{e}\,\text{=}\,\text{2}\text{.71828}...$ is the base of the natural logarithm and $\gamma \,=\,0.57721...$ is the usual Euler–Mascheroni constant. This strengthens a recent theorem of Olofsson.

Keywords

11N8011N0511M45Beurling primeMertens’ theoremgeneralized primearithmetic semigroupabstract analytic number theory
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 4 , 01 December 2013 , pp. 829 - 843
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Beurling, A., Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. I. Acta Math. 68 (1937), no. 1, 255–291.Google Scholar
[2] Diamond, H. G., A set of generalized numbers showing Beurling's theorem to be sharp. Illinois J. Math. 14 (1970), 29–34.Google Scholar
[3] Diamond, H. G., Chebyshev estimates for Beurling generalized prime numbers. Proc. Amer. Math. Soc. 39 (1973), 503–508. http://dx.doi.org/10.1090/S0002-9939-1973-0314782-4 Google Scholar
[4] Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series, and products. Seventh ed., Elsevier/Academic Press, Amsterdam, 2007.Google Scholar
[5] Hall, R. S., Beurling generalized prime number systems in which the Chebyshev inequalities fail. Proc. Amer. Math. Soc. 40 (1973), 79–82. http://dx.doi.org/10.1090/S0002-9939-1973-0318085-3 Google Scholar
[6] Korevaar, J., Tauberian theory: A century of developments. Grundlehren der Mathematischen Wissenschaften, 329, Springer-Verlag, Berlin, 2004.Google Scholar
[7] Lebacque, P., Generalised Mertens and Brauer-Siegel theorems. Acta Arith. 130 (2007), no. 4, 333–350. http://dx.doi.org/10.4064/aa130-4-3 Google Scholar
[8] Olofsson, R., Properties of the Beurling generalized primes. J. Number Theor. 131 (2011), no. 1, 45–58. http://dx.doi.org/10.1016/j.jnt.2010.06.014 Google Scholar
[9] Parry, W. and Pollicott, M., Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990), 268 pp.Google Scholar
[10] Pollicott, M., Agmon's complex Tauberian theorem and closed orbits for hyperbolic and geodesic flows. Proc. Amer. Math. Soc. 114 (1992), no. 4, 1105–1108.Google Scholar
[11] Pollicott, M., Periodic orbits and zeta functions. In: Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 409–452.Google Scholar
[12] Rosen, M., A generalization of Mertens’ theorem. J. Ramanujan Math. Soc. 14 (1999), no. 1, 1–19.Google Scholar
[13] Sharp, R., An analogue of Mertens’ theorem for closed orbits of Axiom A flows. Bol. Soc. Brasil. Mat. (N.S.. 21 (1991), no. 2, 205–229. http://dx.doi.org/10.1007/BF01237365 Google Scholar
[14] Zhang, W.-B., Density and O-density of Beurling generalized integers. J. Number Theor. 30 (1988), no. 2, 120–139. http://dx.doi.org/10.1016/0022-314X(88)90012-1 Google Scholar