#### 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 ${\sum }_{|h|, where $\tau$ denotes the divisor function and $h\in ℤ\setminus \left\{0\right\}$. We consider in particular the special cases where $f$ is the generalized divisor function ${\tau }_{z}$ 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|, where $\omega \left(n\right)$ 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 ${\tau }_{\alpha }$ with $\alpha \in ℚ$, and an interpolation argument in the $z$-variable for weighted mean values of ${\tau }_{z}$. The second is based on an identity of Linnik type for ${\tau }_{z}$ and the well-factorability of friable numbers.

##### Keywords
shifted convolution, divisor function, combinatorial identity
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