On the distribution of values of Hardy’s Z-functions in short intervals II: The q-aspect
Moscow Journal of Combinatorics and Number Theory
We continue our investigations regarding the distribution of positive and negative values of Hardy’s Zfunctions 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 1\2+ 1\2 η < ϑ ≤ 1, the Lebesgue measure of the set of 2 values of t ϵ [T, T + H] for which Z(t, χ) > 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.
Mawia, Ramdin, "On the distribution of values of Hardy’s Z-functions in short intervals II: The q-aspect" (2019). Journal Articles. 994.