Lower estimates related to the twisted Dedekind function

Abstract

Let χ (mod q) be a Dirichlet character. The main goal of this paper is to study oscillations of the differencewhere ψ χ (n) = n Q p|n (1+χ(p)/p) denotes the twisted Dedekind function. We prove that for infinitely many odd characters χ called "good", we have E(x, χ) = Ω(x log log x), and E(x, χ) = Ω ± (x log log x) when χ is real. We give a necessary and sufficient condition for χ to be good, and in particular we prove that all odd primitive characters are good. We show also that there are infinitely many moduli q ≥ 3, including all prime powers q ≥ 3, for which all odd characters χ (mod q) are good.