Moscow Journal of Combinatorics and Number Theory

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

Ramdin Mawia

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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 qTη, H=Tϑ, with ϑ>0, η>0 satisfying 12+12η<ϑ1, 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.

Article information

Mosc. J. Comb. Number Theory, Volume 8, Number 3 (2019), 229-245.

Received: 10 November 2018
Revised: 7 May 2019
Accepted: 31 May 2019
First available in Project Euclid: 13 August 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses

Hardy's function Hardy–Selberg function Dirichlet $L$-function value distribution


Mawia, Ramdin. On the distribution of values of Hardy's $Z$-functions in short intervals, II: The $q$-aspect. Mosc. J. Comb. Number Theory 8 (2019), no. 3, 229--245. doi:10.2140/moscow.2019.8.229.

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