On the zeros of the Lerch zeta-function. II (Q2769677)

In the paper under review various results on the zero-distribution of the Lerch zeta-function are obtained. [Part I, cf. Liet. Mat. Rink. 41, Special Issue, 53-57 (2001)]. For \(\Re s>1\) the Lerch zeta-function is given by NEWLINE\[NEWLINE L(\lambda,\alpha,s)=\sum_{n=0}^\infty (n+\alpha)^{-s}\exp(2\pi i\lambda n),NEWLINE\]NEWLINE and by analytic continuation elsewhere; here \(\lambda\) and \(\alpha\) are real parameters with \(0<\alpha\leq 1\). Denote by \(A_T(\lambda,\alpha;a,b)\) the following assertion: for any \(\sigma_1,\sigma_2\), satisfying \(a<\sigma_1<\sigma_2<b\), there exists a positive constant \(c\), depending on \(\lambda,\alpha,\sigma_1\) and \(\sigma_2\), such that \(L(\lambda,\alpha,s)\) has for sufficiently large \(T\) more than \(cT\) zeros in the rectangle \(\sigma_1<\Re s<\sigma_2, |\text{Im }s|<T\). Extending some classical results of \textit{H. Davenport} and \textit{H. Heilbronn} [J. Lond. Math. Soc. 11, 181-185, 307-312 (1936; Zbl 0014.21601, Zbl 0015.19802)] and \textit{J. W. S. Cassels} [J. Lond. Math. Soc. 36, 177-184 (1961; Zbl 0097.03403)] for Hurwitz zeta-functions \(L(0,\alpha,s)\), the author proves among other results: NEWLINENEWLINENEWLINEi) if \(\alpha\) is irrational, then there exists a constant \(\delta\in(0,\alpha)\), depending on \(\lambda\) and \(\alpha\), such that \(A_T(\lambda, \alpha;1,1+\delta)\) is true; NEWLINENEWLINENEWLINEii) if \(\alpha\) is transcendental, then the assertion \(A_T(\lambda,\alpha; 1/2,1)\) holds. NEWLINENEWLINENEWLINEThe proof of the latter result relies on \textit{A. Laurinčikas'} universality theorem [Lith. Math. J. 37, 275-280 (1997; Zbl 0938.11045)]. Further, by a nice continuity argument, some zero-free regions for Lerch zeta-functions are obtained.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].