The universality of Dirichlet series attached to finite Abelian groups (Q2710124)

In this interesting paper the author proves the universality of a Dirichlet series associated to finite Abelian groups. NEWLINENEWLINENEWLINEFor positive integers \(m\) let \(t_2(m)\) be the sum of the numbers of subgroups of all Abelian groups \(G\) of order \(m\) and of rank at most \(2\). It is well known that the generating Dirichlet series, defined by \(H(s)=\sum_{m=1}^\infty t_2(m)m^{-s}\), has the Euler product representation NEWLINE\[NEWLINE H(s)=\zeta^2(s)\zeta^2(2s)\zeta(2s-1)\prod_p(1+p^{-2s}-2p^{-3s}),NEWLINE\]NEWLINE where the product is taken over all prime numbers, and \(\zeta(s)\) is the Riemann zeta-function [see \textit{G. Bhowmik} and \textit{O. Ramaré}, Acta Arith. 66, 45-62 (1994; Zbl 0795.11047)]. In [Publ. Math. 52, 517-533 (1998; Zbl 0923.11124)] the author proved a limit theorem in the sense of weak convergence of probability measures for \(H(s)\). This together with a denseness result in the space of analytic functions on \(D=\{s : 3/4<\text{Re }s<1\}\) proves the universality theorem: NEWLINENEWLINENEWLINELet \(K\) be a compact subset of the strip \(D\) with connected complement. Let \(f(s)\) be a non-vanishing continuous function on \(K\) which is analytic in the interior of \(K\). Then for any \(\varepsilon>0\) NEWLINE\[NEWLINE \liminf_{T\to\infty}{1\over T}\text{meas}\left\{\tau\in[0,T] : \sup_{s\in K}|H(s+i\tau)-f(s)|<\varepsilon\right\}>0.NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].