Multiplicative functions that are close to their mean

Abstract

We introduce a simple approach to study partial sums of multiplicative functions which are close to their mean value. As a first application, we show that a completely multiplicative function f :with c = 0 if and only if f (p) = 1 for all but finitely many primes and |f (p)| < 1 for the remaining primes. This answers a question of Imre Ruzsa.For the case c = 0, we show, under the additional hypothesis p:|f (p)|<1 1/p < ∞, that f has bounded partial sums if and only if f (p) = χ(p)p it for some non-principal Dirichlet character χ modulo q and t ∈ R except on a finite set of primes that contains the primes dividing q, wherein |f (p)| < 1. This essentially resolves another problem of Ruzsa and generalizes previous work of the first and the second author on Chudakov's conjecture.We also consider some other applications, which include a proof of a recent conjecture of Aymone concerning the discrepancy of square-free supported multiplicative functions.

The Erdős discrepancy problem over the squarefree and cubefree integers

“…Some partial results in the direction of Theorem 1.1 were previously obtained by the author in a preprint version of this paper, and later by Klurman et al. [6] in their preprint (arXiv, version 1). In a newer version of [6] (to appear in Trans.…”

Beyond the Erdős discrepancy problem in function fields

“…By an argument similar to that surrounding (10), this implies that χ 2 is principal and ξ 2 ≡ 1. Furthermore, in this case, as in [5, p. 15] we have…”

“…This contradicts (10). It must follow that ξ j = ξ j+1 for all j sufficiently large (in terms of C).…”

“…To accomplish this reduction we use a technical device, which we call the "rotation trick." This general trick played a crucial role in our recent work [10] on multiplicative functions over the integers. After this reduction to f satisfying (6), it remains to treat the case where f is a modified character (see Definition 1.6 above).…”

“…By means of the following proposition, however, we can in fact restrict ourselves to functions differing from a twisted character at a bounded number of irreducibles, only. The proof of the proposition utilizes what we call a "rotation trick"; see [10] for applications of the same idea in the integer setting.…”