On certain arithmetic functions \(\tilde M(s; z_{1}; z_{2})\) associated with global fields: analytic properties (Q533374)

In the paper, the analytic properties of the certain arithmetic functions \({\tilde M}(s;z_1,z_2)\) associated with global fields \(K\) are discussed. The author studies the general density functions \(M(s)| dx|\) on \(\mathbb R^d\), \(d=1,2,\dots\), with center 0, in particular, the best lower bound for the quantity \(\mu^{d/2}\nu\) (here \(\mu\) is the variance, \(\nu\) is the Plancherel volume). One of results is devoted to the investigation of the limit of \(\mu_\sigma\nu_\sigma\) and \(\mu_\sigma M_\sigma(\mu_\sigma^{1/2}w)\) as \(\sigma \to 1/2, +\infty\). The analytic continuation to the space \(\{\text{Re}\,s >0\}\setminus\{\rho/(2n); n \in \mathbb N, \zeta(\rho=0 \;\text{or} \;\infty)\}\times\mathbb C^2\) of the function \({\tilde M}(s;z_1,z_2)\) is proved (here \(\zeta(s)\) is the zeta-function of \(K\) without factors of finite set of prime divisors of \(K\) including all archimedean primes in the number field case). The last part of the paper is devoted to the studies of the rapid decay property of \(|{\tilde M}_\sigma(z)|^2\) with attention when \(\sigma\) is arbitrarily close to \(1/2\) and \(| z|\) is unbounded.