On a problem of Karatsuba
type: Item
instance of: scholarly article
title: On a problem of Karatsuba
zbMATH Open document ID: 1250.11085
DOI: 10.1134/S000143461009021X
author: S. A. Gritsenko
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: In [``On the zeros of the Davenport-Heilbronn function lying on the critical line'', Math. USSR, Izv. 36, No. 2, 311--324 (1991); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 2, 303--315 (1990; Zbl 0717.11035)] and [``A new approach to the problem of the zeros of some Dirichlet series,'' Proc. Steklov Inst. Math. 207, 163--177 (1995); translation from Tr. Mat. Inst. Steklova 207, 180--196 (1994; Zbl 0904.11024)] \textit{A. A. Karatsuba} developed and refined a method for finding lower bounds for the number of zeros of certain Dirichlet series on the intervals of the critical line, based on the lower bounds for the fractional moments of the series on the critical line. Let \(\chi\) be a Dirichlet character, \(\tau(\chi)\) its Gauss sum; define \(\varepsilon(\chi):= (i^{\delta} \sqrt{k})/\tau(\chi)\), where \(k\) is the modulus of \(\chi\) and \(\delta\) is equal to 0 or 1, respectively, depending on whether \(\chi\) is even or odd. Let \[ \rho(s, \chi):= \varepsilon(\overline{\chi})\left( \frac{\pi}{k}\right)^{s-1/2} \frac{\Gamma((1-s+\delta)/2)}{\Gamma((s+\delta)/2)} \] and define \[ \Phi(s)=a_1(\rho(s,\chi_1))^{-1/2}L(s,\chi_1)+a_2(\rho(s,\chi_2))^{-1/2}L(s,\chi_2), \] for arbitrary real numbers \(a_1\), \(a_2\). In the paper under review order-sharp estimates of fractional moments of function \(G(t):=\Phi(1/2 + it)\) (an analogue of the Hardy function) are derived and the lower bound \(cT (\log T) ^{1/6}\) for the number of zeros of \(G(t)\) with \(t\in [0,T]\) is deduced, under some additional assumptions on numbers \(a_1\), \(a_2\) and moduli of characters \(\chi_1\) and \(\chi_2\). For certain choices of characters \(\chi_1\) and \(\chi_2\) this lower bound represents an improvement of Karatsuba's lower bound (op.cit.).
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 5980691
zbMATH Keywords: Fractional moments of the Hardy function
zbMATH Keywords: Dirichlet L-function
zbMATH Keywords: lower bounds for number of zeros on the intervals of the critical line
reviewed by: Lejla Smajlovic
MaRDI profile type: Publication
cites work: Q3882540
cites work: Q3478470
cites work: Q3996835
cites work: Q3363327
cites work: On fractional moments of Dirichlet \(L\)-functions. II
cites work: Fractional Moments of the Riemann Zeta-Function
Recommended article: Q5239668
Recommended article: Fractional moments and zeros of \(\zeta(s)\) on the critical line
Recommended article: Q3478470
Recommended article: Q4397625
Recommended article: On the connection of fractional moments of the Riemann zeta-function with its zeros
Recommended article: Fractional moments of the \(\zeta\)-function
Recommended article: On the distribution of zeros of linear combinations of Dirichlet \(L\)-functions on the critical line
Recommended article: Q4029822
Recommended article: Q5385698
Recommended article: Q5239747
description: scientific article; zbMATH DE number 5980691