Combinatorial identities and Titchmarsh's divisor problem for multiplicative functions

TL;DR

This paper derives full asymptotic expansions for shifted convolution sums involving multiplicative functions, extending classical divisor problems and employing combinatorial and factorization techniques.

Contribution

It provides new asymptotic formulas for convolution sums with multiplicative functions, including special cases like sums of two squares and generalized divisor functions.

Findings

Asymptotic expansion for sums involving $ au(n-h)$ and multiplicative functions.

Extension of Titchmarsh divisor problem to new settings.

Two different proofs using combinatorial and factorization methods.

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.