On statistical properties of the Lerch zeta-function (Q1873226)

The Lerch zeta-function with parameters \(0<\lambda,\alpha\leq 1\) is given for \(\Re s>1\) by the Dirichlet series \[ L(\lambda,\alpha,s)=\sum_{n=0}^\infty {\exp(2\pi i\lambda)\over (n+\alpha)^s}, \] and by analytic continuation elsewhere except for at most one simple pole at \(s=1\). Being a generalization of the famous Riemann zeta-function \(\zeta(s)=L(1,1,s)\), the value-distribution of the Lerch zeta-function with respect to the parameters \(\lambda,\alpha\) is of special interest. In the paper under review the author proves a discrete limit theorem (in the sense of weakly convergent probability measures) for \(L(\lambda,\alpha,s)\) on the complex plane under the assumptions that \(\Re s>1/2\) and that \(\alpha\) is transcendental. The first of these assumptions is natural (in view of the existence of the mean square of \(L(\lambda,\alpha,s)\) on vertical lines in the half-plane \(\Re s>1/2\)). The second assumption assures the linear independence of the numbers \(\log (n+\alpha), n=0,1,\ldots\), which is very useful for proving this or allied limit theorems (it is an open problem to prove a continuous or discrete limit theorem in the case of algebraic irrational \(\alpha\)). For Part II, see Zbl 1028.11054 below.