Extreme values of zeta and L-functions

Abstract

We introduce a resonance method to produce large values of the Riemann zeta-function on the critical line, and large and small central values of L-functions.

An arithmetical mapping and applications to Ω-results for the Riemann zeta function

“…The author is grateful to the referee for pointing out the recent paper by Soundararajan [11], in which a "resonator" method was developed and used to find Ω-results for ζ(1/2 + it). The method employed in this paper regarding Ω-results for ζ(α + it) is similar in nature.…”

Arithmetic, Zeros, and Nodal Domains on the Sphere

“…While everything mentioned so far is expected to be true, one cannot push conjectures about Hecke eigenfunctions behaving uniformly like typical random waves too far. By adapting a method of Soundararajan from [28], Milićević has shown in [22] that on arithmetic Riemann surfaces the Hecke eigenfunctions take on values significantly larger than predicted by the RWM for these surfaces. It is likely that this proof goes through on the sphere to give a similar result.…”

“…Supposing there are few φ changing sign between x and y, so that few of the φ(x)φ(y) are negative, we aim to 1. Choose α to kill off all the negative contributions to (28). 2.…”

Gram's law in the theory of the Riemann zeta-function. Part 1

“…[27], [28] и [29]). Менее очевидной оказывается ситуация в случае, когда вместо максимума значений дзета-функции на соответствующем множестве точек критической прямой рассматривается минимум.…”