A note on the zeros of zeta and L-functions

Abstract

Let π S(t) denote the argument of the Riemann zeta-function at the point s = 1 2 + it. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for S(t). We discuss a generalization of this bound for a large class of L-functions including those which arise from cuspidal automorphic representations of GL(m) over a number field. We also prove a number of related results including bounding the order of vanishing of an L-function at the central point and bounding the height of the lowest zero of an L-function.

Pair correlation estimates for the zeros of the zeta function via semidefinite programming

“…Minimizing (or maximizing) the functional over the class A would then produce the best bound one can possibly get with that specific approach. Nowadays, this idea is a standard technique in analytic number theory (introduced first by Beurling and Selberg) and the following are some references (clearly not a complete list) where the main approach is exactly that: Large sieve inequalities [31,32]; Erdös-Turán inequalities [15,40]; Hilbert-type inequalities [12,13,15,30,31,40]; Tauberian theorems [31]; Bounds in the theory of the Riemann zetafunction and L-functions [5,6,7,8,9,10,11,17,19,26,27]; Prime gaps [14]. From our point of view, our main contribution connects here.…”

A Note on Entire L-Functions

“…These results are based on the generalized Riemann hypothesis, which states that Λ(s, π) = 0 if Re s = 1 2 . As in [5,6,8,10], the analytic conductor of L(s, π), which is defined by…”

“…We use this in the representation lemma and finally optimize the support of some Fourier transforms resulting from the previous analysis to get the desired result. We highlight that one of the main technical difficulties of our proof, when compared with results in [5,6,8,10], is in the analysis of the sums over prime powers. To obtain the exact asymptotic behavior of such tough terms we shall need explicit formulas for the Fourier transforms of these extremal functions.…”

“…In particular, we have estimates for the Riemann zeta-function [3,4,5,6,7,10] under the Riemann hypothesis. For a general family of L-functions in the framework of [11,Chapter 5], we have similar estimates in the critical line [5,6,8,10] under the generalized Riemann hypothesis. The purpose of this paper is to exhibit explicit bounds for a family of entire L-functions in the critical strip.…”

Interpolation formulas with derivatives in de Branges spaces II

“…These are functions F of prescribed exponential type that minimize the L 1 (R, µ)-distance (µ is some non-negative measure) from a given function g and such that F lies below or above g on R. These functions have the special property that they interpolate the target function g (an its derivative) at a certain sequence of real points and have several special properties that are very useful in applications to analytic number theory, being the key to provide sharp (or improved) estimates. For instance, in connection to: large sieve inequalities [24,28], Erdös-Turán inequalities [15,28], Hilbert-type inequalities [12,14,15,22,28], Tauberian theorems [22] and bounds in the theory of the Riemann zeta-function and general L-functions [7,8,9,11,16,18,19]. Further constructions and applications can also be found in [6,10,13,21,25].…”