Abstract
Given a multiplicative function which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum , where denotes the divisor function and . We consider in particular the special cases where is the generalized divisor function with , 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 , where counts the number of distinct prime divisors of , 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 with , and an interpolation argument in the -variable for weighted mean values of . The second is based on an identity of Linnik type for and the well-factorability of friable numbers.
Citation
Sary Drappeau. Berke Topacogullari. "Combinatorial identities and Titchmarsh's divisor problem for multiplicative functions." Algebra Number Theory 13 (10) 2383 - 2425, 2019. https://doi.org/10.2140/ant.2019.13.2383
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