Distribution of signs of Karatsuba's and generalized Davenport-Heilbronn Z-functions

JOURNAL OF NUMBER THEORY

卷:212

Article

Das, Mithun Kumar¹  Pujahari, Sudhir² 

[1] Harish Chandra Res Inst HBNI, Chatnag Rd, Allahabad 211019, Uttar Pradesh, India

[2] Univ Hong Kong, Pokfulam, Hong Kong, Peoples R China

关键词: Dirichlet L-series;    Davenport-Heilbronn function;    Hardy's Z-function;    Karatsuba's Z-function;   

DOI  :  10.1016/j.jnt.2019.11.012

来源: Elsevier

【 摘 要 】

For 1 <= i <= r, let chi i be primitive Dirichlet characters modulo qi and Z(t, chi i) be the Z-function corresponding to the Dirichlet L-series L(s, chi i). Let Omega(t) be a real linear combination of Z(t, chi i). Since Z(t, chi i) is real for real t, Omega(t) is real for real t. In this paper, we show that the Lebesgue measure of the set, where the functional values of Omega(t) is positive or negative in the interval [T, 2T] is at least T/r(2) . We also study the Lebesgue measure of the set that the certain complex linear combinations of Z(t, chi i) takes positive or negative values respectively. In particular, we study the distribution of signs of the Z-function correspond to the Davenport-Heilbronn function. Moreover, we prove that for sufficiently large T, the generalized Davenport-Heilbronn function has at least H(log T)(2/phi(q)-is an element of) odd order zeros along the critical line on the interval [T,T + H]. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free