Chebyshev estimates for Beurling generalized prime numbers. I

2012/01/31 by Jasson Vindas · 10 citations
Mathematics · #Analytic Number Theory Research #Chebyshev filter #Combinatorics #Discrete mathematics #Function (biology) #History and Theory of Mathematics #Limits and Structures in Graph Theory #Mathematical analysis #Mathematics #Prime (order theory) #Prime number #Prime number theorem #math.NT #msc:11M41 #msc:11N05 #msc:11N80

paper · pdf · doi:10.1016/j.jnt.2012.04.019

published in Journal of Number Theory 132(10), 2371-2376 (Elsevier BV) · 6 pages

arxiv created 2012/06/04 · openalex publication_date 2012/06/26 · arxiv updated 2013/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We provide new sufficient conditions for Chebyshev estimates for Beurling generalized primes. It is shown that if the counting function N of a generalized number system satisfies the L¹-condition ∫₁^∞|(N(x)-ax)/(x)|(dx)/(x)<∞ and N(x)=ax+o(x/log x), for some a>0, then 0<\liminf_x→∞(ψ(x))/(x) and \limsup_x→∞(ψ(x))/(x)<∞ hold. We give an analytic proof of this result. It is based on Wiener division theorem. Our result extends those of Diamond (Proc. Amer. Math. Soc. 39 (1973), 503--508) and Zhang (Proc. Amer. Math. Soc. 101 (1987), 205--212).

Citations

• Tauberian Theory