The space of Dirichlet series to multivariate lattices

Abstract

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 𝜃-power density of a Dirichlet series is introduced. A connection is established between the 𝐶 𝜃-power density of the Dirichlet series and the abscissa of its absolute convergence.
It is established that if the Dirichlet series 𝑓(𝛼|Λ) satisfies the conditions of the generalized Selberg lemma with 𝜃1 < 𝜃, then the Dirichlet series 𝑓(𝛼|Λ) extends analytically into the halfplane
with 𝑠𝑖𝑔𝑚𝑎 > 𝜃1, except for the point 𝛼 = 𝜃, at which it has a first-order pole with a subtraction of 𝐶𝜃.
A new concept 𝐶 logarithmic 𝜃-power density of the Dirichlet series is introduced. It has been established that if the Dirichlet series 𝑓(𝛼|Λ) has 𝐶 logarithmic 𝜃-power density and 𝜃 < 1, then the abscissa of absolute convergence holds the equality 𝜎𝑓 = 0 and The Dirichlet series 𝑓(𝛼|Λ) is a holomorphic function in the entire right 𝛼-half-plane with 𝜎 > 0.
It is shown that if the Dirichlet series 𝑓(𝛼|Λ) has 𝐶 logarithmic 𝜃-power density and 𝜃 < 1, then The holomorphic domain of the zeta function 𝜁(𝑀|𝛼) is 𝛼-the half-plane 𝜎 > 0.