The space of Dirichlet series to multivariate lattices

N. V. Maksimenko ; I. Yu. Rebrova

Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 206-211

Résumé

The work considers the set of all possible Dirichlet series generated by a given lattice, and studies the properties of this function space over the field of complex numbers. A new concept of $C$ $\theta$-power density of a Dirichlet series is introduced. A connection is established between the $C$ $\theta$-power density of the Dirichlet series and the abscissa of its absolute convergence. It is established that if the Dirichlet series $f(\alpha|\Lambda)$ satisfies the conditions of the generalized Selberg lemma with $\theta_1\theta$, then the Dirichlet series $f(\alpha|\Lambda)$ extends analytically into the half-plane with $\ sigma>\theta_1$, except for the point $\alpha=\theta$, at which it has a first-order pole with a subtraction of $C\theta$. A new concept $C$ logarithmic $\theta$-power density of the Dirichlet series is introduced. It has been established that if the Dirichlet series $f(\alpha|\Lambda)$ has $C$ logarithmic $\theta$-power density and $\theta1$, then the abscissa of absolute convergence holds the equality $\sigma_f=0$ and The Dirichlet series $f(\alpha|\Lambda)$ is a holomorphic function in the entire right $\alpha$-half-plane with $\sigma>0$. It is shown that if the Dirichlet series $f(\alpha|\Lambda)$ has $C$ logarithmic $\theta$-power density and $\theta1$, then The holomorphic domain of the zeta function $\zeta(M|\alpha)$ is $\alpha$-the half-plane $\sigma>0$.

Détail

Keywords: Riemann zeta function, Dirichlet series, zeta function of the monoid of natural numbers.