Combinatorial identities and Titchmarsh's divisor problem for multiplicative functions

Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum $\sum_{|h|<n\leq x} f(n) τ(n-h)$, where $τ$ 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 $τ_z$ 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 $\sum_{|h|<n\leq x,\,ω(n)=k} τ(n-h)$, where $ω(n)$ 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 $τ_α$ with $α\in\mathbb{Q}$, and an interpolation argument in the $z$-variable for weighted mean values of $τ_z$. The second is based on an identity of Linnik type for $τ_z$ and the well-factorability of friable numbers.

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