Approximants for classical arithmetic functions

Abstract: Many classical arithmetic functions such as the M\"obius function \mu(n), the von Mangoldt function \Lambda(n), or the higher order divisor functions d_k(n) are notoriously difficult to work with: for instance obtaining cancellation for \sum_{n \leq x} \mu(n) \mu(n+1) is part of the Chowla conjecture, obtaining an asymptotic for \sum_{n \leq x} \Lambda(n) \Lambda(n+2) would give the twin prime conjecture, and even guessing the full main term expansion for \sum_{n \leq x} d_k(n) d_l(n+1) is a non-trivial task (and verifying it is still open when k,l > 2). However, in all these cases one can propose _approximants_ \mu^\sharp, \Lambda^\sharp, d_k^\sharp to these functions that are substantially easier to work with (mostly by virtue of being "Type I sums") and which are (either rigorously or heuristically) close to the original functions \mu, \Lambda, d_k in various useful ways. We present recent and forthcoming work with Ter\"av\"ainen, Matom\"aki--Shao--Ter\"av\"ainen, and Matom\"aki--Radziwi{\l}{\l}--Shao--Ter\"av\"ainen using these approximants to control Gowers uniformity norms and related statistics for these functions, as well as to verify cases of a unified Hardy-Littlewood-Chowla conjecture in the presence of a Siegel zero.

number theory

Audience: researchers in the topic

CRM-CICMA Québec Vermont Seminar Series

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