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 |h|<n≤x f (n)τ (n − h), where τ denotes the divisor function and h ∈ Z {0}. We consider in particular the special cases where f is the generalized divisor function τz with z ∈ 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 |h|<n≤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 τα with α ∈ Q, and an interpolation argument in the z-variable for weighted mean values of τz. The second is based on an identity of Linnik type for τz and the well-factorability of friable numbers.

Multiplicative functions in large arithmetic progressions and applications

“…As a consequence, we obtain an alternative proof of the results of [5] briefly described in Section 2 and, in a more innovative way, we obtain new information on the joint distribution of (f (n), g(n − 1)), for certain additive functions f , g.…”

“…where the λ h,j are entire functions. Note that the error term above is actually slightly more precise than stated in [5]. Let S denote the left-hand side of (2.1).…”

“…We list here, among many possible ones, four applications of our main results. The first is a quick, natural proof of theorems 1.3, 1.4 and 1.6 of [5] and their corollaries.…”

“…The idea of this paper came up to our minds on studying the work of Drappeau and Topacogullari [5], in which the authors investigate sums of the form…”

The Thue-Morse and Rudin-Shapiro sequences at primes in principal number fields

“…One inconvenient, as with all methods which pass through the von Mangoldt function, is the necessity to use partial summation to detect the size of log N (n). Here we take the opportunity to proceed along a slightly different argument (see [12,Theorem 3.3]), with the benefit that we avoid completely partial summation.…”