Vol. 8, No. 3, 2019

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

Ramdin Mawia

Vol. 8 (2019), No. 3, 229–245

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 1 2 + 1 2η < ϑ 1, the Lebesgue measure of the set of values of t [T,T + H] for which Z(t,χ) > 0 is (φ(q)24ω(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.

Hardy's function, Hardy–Selberg function, Dirichlet $L$-function, value distribution
Mathematical Subject Classification 2010
Primary: 11M06, 11M26
Received: 10 November 2018
Revised: 7 May 2019
Accepted: 31 May 2019
Published: 23 July 2019
Ramdin Mawia
Theoretical Statistics and Mathematics Unit
Indian Statistical Institute