Almost periodicity and large oscillations of prime counting functions

Speaker: Jan-Christoph Schlage-Puchta

Date: Fri, Jun 21, 2024

Location: PIMS, University of British Columbia

Conference: Comparative Prime Number Theory

Subject: Mathematics, Number Theory

Class: Scientific

CRG: L-Functions in Analytic Number Theory


If we assume the relevant Riemann hypotheses, after a suitable rescaling many functions counting certain primes become almost periodic. There are different notion of almost periodicity in use; here we consider the notion induced by the norm $||f|| = \sup_{x∈\mathbb{R}} \int_x^{x+1} |f(t)|^2\,dt$. We show that if a function $f$ can be approximated by linear combinations of periodic functions with respect to this norm, then the level sets $\left\{x: f(x) \geq t\right\}$ are almost periodic for all real $t$ with at most countably many exceptions. We also compare this notion to other notions of almost periodicity in use.

Please note, the wrong video feed was captured for this lecture so the writing on the blackboard is not legible.