Vol. 13, No. 10, 2019

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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
Abstract

Given a multiplicative function f which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum |h|<nxf(n)τ(n h), where τ denotes the divisor function and h {0}. We consider in particular the special cases where f is the generalized divisor function τz with z , 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 |h|<nx,ω(n)=kτ(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 τα with α , and an interpolation argument in the z-variable for weighted mean values of τz. The second is based on an identity of Linnik type for τz and the well-factorability of friable numbers.

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