2014/02/26 by Youness Lamzouri, Lamzouri, Youness, Stephen Lester +3
Mathematics · #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1402.6682
openalex publication_date 2014/02/26 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28
We investigate the distribution of the Riemann zeta-function on the line\n Re(s)=\σ. For tfrac 12 < \σ \≤ 1 we obtain an upper bound on the\ndiscrepancy between the distribution of \ζ(s) and that of its random\nmodel, improving results of Harman and Matsumoto. Additionally, we examine the\ndistribution of the extreme values of \ζ(s) inside of the critical strip,\nstrengthening a previous result of the first author.\n As an application of these results we obtain the first effective error term\nfor the number of solutions to \ζ(s) = a in a strip tfrac12 < \σ1 <\n\σ2 < 1. Previously in the strip tfrac 12 < \σ < 1 only an\nasymptotic estimate was available due to a result of Borchsenius and Jessen\nfrom 1948 and effective estimates were known only slightly to the left of the\nhalf-line, under the Riemann hypothesis (due to Selberg) and to the right of\nthe abscissa of absolute convergence (due to Matsumoto). In general our results\nare an improvement of the classical Bohr-Jessen framework and are also\napplicable to counting the zeros of the Epstein zeta-function.\n