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 $\sum_{|h|<n\leq x} f(n) \tau(n-h)$, where $\tau$ denotes the divisor function and $h\in\mathbb{Z}\setminus\{0\}$. We consider in particular the special cases where $f$ is the generalized divisor function $\tau_z$ with $z\in\mathbb{C}$, 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 $\sum_{|h|<n\leq x,\,\omega(n)=k} \tau(n-h)$, where $\omega(n)$ counts the number of distinct prime divisors of $n$, thus extending a result of Fouvry and Bombieri-Friedlander-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.