Discrepancy bounds for the distribution of the Riemann zeta-function and applications

Abstract

We investigate the distribution of the Riemann zeta-function on the line Re(s) = σ. For 1 2 < 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 1 2 < σ 1 < σ 2 < 1. Previously in the strip 1 2 < 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) and to the right of the abscissa of absolute convergence (due to Matsumoto). 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.

The Density Function for the Value-Distribution of the Lerch Zeta-Function and Its Applications

“…We prove the following estimate in this section. The proof is based on the method in Section 5 of [14]. Proposition 6.2.…”

On the zeros of Epstein zeta functions near the critical line

“…holds for any 1/2 < σ 1 ≤ σ 2 and some nonnegative continuous function g(σ). By a straightforward adaptation of [7], the author in [9] improved the above asymptotic formula to…”

The a-values of the Riemann zeta function near the critical line

“…Recently, Lamzouri, Lester and Radziwiłł [7] reduced the size of the error in (1) and proved that N a (σ 1 , σ 2 ; T ) = c(a, σ 1 , σ 2 )T + O T log log T (log T ) σ 1 /2 holds for fixed 1/2 < σ 1 < σ 2 < 1 and T ≥ 3.…”

“…is small for σ = σ T , where the supremum is taken over all rectangular regions in the complex plane with their sides parallel to real or imaginary axis. Bounding the discrepancy has been studied by various authors (e.g., [5], [7]). In particular, Lamzouri et al [7,Theorem 1.1] showed that…”

“…Bounding the discrepancy has been studied by various authors (e.g., [5], [7]). In particular, Lamzouri et al [7,Theorem 1.1] showed that…”