Combinatorial identities and Titchmarsh’s divisor problem for multiplicative functions

Drappeau, Sary; Topacogullari, Berke

doi:10.2140/ant.2019.13.2383

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Combinatorial identities and Titchmarsh's divisor problem for multiplicative functions

Sary Drappeau and Berke Topacogullari

Vol. 13 (2019), No. 10, 2383–2425

DOI: 10.2140/ant.2019.13.2383

Abstract

Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum $\sum_{| h | < n \leq x} f (n) τ (n - h)$ , where $τ$ denotes the divisor function and $h \in ℤ ∖ {0}$ . We consider in particular the special cases where $f$ is the generalized divisor function $τ_{z}$ with $z \in ℂ$ , and the characteristic function of sums of two squares (or more generally, ideal norms of abelian extensions). As another application, we deduce a full asymptotic expansion in the generalized Titchmarsh divisor problem $\sum_{| h | < n \leq x, ω (n) = k} τ (n - h)$ , where $ω (n)$ counts the number of distinct prime divisors of $n$ , thus extending a result of Fouvry and Bombieri, Friedlander and Iwaniec.

We present two different proofs: The first relies on an effective combinatorial formula of Heath-Brown’s type for the divisor function $τ_{α}$ with $α \in ℚ$ , and an interpolation argument in the $z$ -variable for weighted mean values of $τ_{z}$ . The second is based on an identity of Linnik type for $τ_{z}$ and the well-factorability of friable numbers.

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Keywords

shifted convolution, divisor function, combinatorial identity

Mathematical Subject Classification 2010

Primary: 11N37

Secondary: 11N25

Milestones

Received: 17 December 2018

Revised: 1 July 2019

Accepted: 31 July 2019

Published: 6 January 2020

Authors

Sary Drappeau
Institut de Mathématiques de Marseille Aix-Marseille Université, CNRS, Centrale Marseille Faculté des sciences de Luminy Marseille France

Berke Topacogullari
EPFL SB MATH TAN Station 8 1015 Lausanne Switzerland

Vol. 13, No. 10, 2019