Dirichlet series algebra of a monoid of natural numbers

N. N. Dobrovol'skii ; M. N. Dobrovol'skii ; N. M. Dobrovol'skii ; I. N. Balaba ; I. Yu. Rebrova

Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 180-196

Résumé

In this paper, for an arbitrary monoid of natural numbers, the foundations of the Dirichlet series algebra are constructed either over a numerical field or over a ring of integers of an algebraic numerical field. For any numerical field $\mathbb{K}$, it is shown that the set $\mathbb{D}^*(M)_{\mathbb{K}}$ of all reversible Dirichlet series of $\mathbb{D}(M)_{\mathbb{K}}$ is an infinite Abelian group consisting of series whose first coefficient is nonzero. We introduce the notion of an integer Dirichlet monoid of natural numbers that form an algebra over a ring of algebraic integers $\mathbb{Z}_\mathbb{K}$ of the algebraic field $\mathbb{K}$. It is shown that for a group $\mathbb{U}_\mathbb{K}$ of algebraic units of the ring of algebraic integers of $\mathbb{Z}_\mathbb{K}$ an algebraic field $\mathbb{K}$ the set of $\mathbb{D}(M)_{\mathbb{U}_\mathbb{K}}$ of entire Dirichlet series, $a(1)\in\mathbb{U}_\mathbb{K}$, is multiplicative group. For any Dirichlet series from the Dirichlet series algebra of a monoid of natural numbers, the reduced series, the irreversible part and the additional series are determined. A formula for decomposition of an arbitrary Dirichlet series into the product of the reduced series and a construction of an irreversible part and an additional series is found. For any monoid of natural numbers allocated to the algebra of Dirichlet series, convergent in the entire complex domain. The Dirichlet series algebra with a given half-plane of absolute convergence is also constructed. It is shown that for any nontrivial monoid $M$ and for any real $\sigma_0$, there is an infinite set of Dirichlet series of $\mathbb{D}(M)$ such that the domain of their holomorphism is $\alpha$-half-plane $\sigma>\sigma_0$. With the help of the universality theorem S. M. Voronin managed to prove the weak form of the universality theorem for a wide class of Zeta functions of monoids of natural numbers. In conclusion describes the actual problem with the Zeta functions of monoids of natural numbers that require further research. In particular, if the Linnik–Ibrahimov hypothesis is true, then a strong theorem of universality should be valid for them.

Détail

Keywords: Riemann zeta function, Dirichlet series, zeta function of the monoid of natural numbers, Euler product, universality theorem, Dirichlet series algebra.