Discrepancy bounds for the distribution of the Riemann zeta-function and applications
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We investigate the distribution of the Riemann zeta-function on the line Re(s) = σ. For ½ < σ ≤ 1 we obtain an upper bound on the discrepancy between the distribution of ζ (s) and that of its random model, improving results of Harman and Matsumoto. Additionally, we examine the distribution of the extreme values of ζ (s) inside of the critical strip, strengthening a previous result of the first author.
As an application of these results we obtain the first effective error term for the number of solutions to ζ (s) = a in a strip ½ < σ1 < σ2 < 1. Previously in the strip ½ < σ< 1 only an asymptotic estimate was available due to a result of Borchsenius and Jessen from 1948 and effective estimates were known only slightly to the left of the half-line, under the Riemann hypothesis (due to Selberg). In general our results are an improvement of the classical Bohr–Jessen framework and are also applicable to counting the zeros of the Epstein zeta-function
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- A. Granville, and K. Soundararajan, Extreme values of | (1 + it)|, in The Riemann Zeta Function and Related Themes: Papers in Honour of Professor K. Ramachandra, Ramanujan Mathematical Society, Mysore, 2006, pp. 65–80.Google Scholar
- Y. Lamzouri, The two-dimensional distribution of values of ζ (1 + it), Int. Math. Res. Not. IMRN (2008), Art. ID rnn 106, 48 pp.Google Scholar
- Y. Lamzouri, On the distribution of extreme values of zeta and L-functions in the strip 12 < s < χ 1, Int. Math. Res. Not. IMRN (2011), 5449–5503.Google Scholar
- K. Matsumoto, Discrepancy estimates for the value-distribution of the Riemann zeta-function. II, in Number Theory and Combinatorics, World Scientific, Singapore, 1985, pp. 265–278.Google Scholar
- A. Selberg, Old and new conjectures and results about a class of Dirichlet series, in Proceedings of the Amalfi Conference on Analytic Number Theory, University of Salerno, Salerno, 1992, pp. 367–385.Google Scholar
- Kai-Man Tsang, The Distribution of the Values of the Riemann Zeta-Function, Ph.D. Thesis, Princeton University, ProQuest LLC, Ann Arbor, MI, 1984.Google Scholar