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 ${\sum }_{\mathrm{\mid }h\mathrm{\mid }<n\le x}f(n)\tau (n-h)$, 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 ${\sum }_{\mathrm{\mid }h\mathrm{\mid }<n\le x,\omega (n)=k}\tau (n-h)$, 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.