Combinatorial identities and Titchmarsh's divisor problem for multiplicative functions

Abstract

Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum ${\sum }_{\left|h\right|<n\le x}f\left(n\right)\tau \left(n-h\right)$, where $\tau$ denotes the divisor function and $h\in \mathbb{Z}\setminus \left\{0\right\}$. 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 ${\sum }_{\left|h\right|<n\le x,\omega \left(n\right)=k}\tau \left(n-h\right)$, 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 $\alpha \in \mathbb{Q}$, 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.