On a problem of Karatsuba (Q650308)

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.).