On the distribution of values of Hardy’s Z-functions in short intervals, II : The q-aspect

Abstract

We continue our investigations regarding the distribution of positive and negative values of Hardy’s Z-functions Z(t,χ) in the interval [T,T+H] when the conductor q and T both tend to infinity. We show that for q≤Tη, H=Tϑ, with ϑ>0, η>0 satisfying 12+12η 0 is ≫(φ(q)2∕4ω(q)q2)H as T→∞, where ω(q) denotes the number of distinct prime factors of the conductor q of the character χ, and φ is the usual Euler totient. This improves upon our earlier result. We also include a corrigendum for the first part of this article.