Convergence domains of the zeta function of some monoids of natural numbers

Abstract

The paper investigates the question of the domain of absolute convergence of the zeta series for some monoids of natural numbers. Two main cases are considered: monoids with 𝐶 power𝜃- density and monoids with 𝐶-logarithmic 𝜃-power density. A new concept is introduced — strong ⃗𝐶= (𝐶1, . . . ,𝐶𝑛) power ⃗𝜃 is the density. For the zeta function of a sequence of natural numbers 𝐴 with a strong ⃗𝐶
= (𝐶1, . . . ,𝐶𝑛)power ⃗𝜃-density proved the theorem according to which the zeta function 𝜁(𝐴|𝛼) is an analytical function of the variable 𝛼, regular at 𝜎 > 0, having 𝑛 poles of the first order, and deductions are found in these poles.
For the case of 𝐶 logarithmic 𝜃-power density, a fundamentally different result is proved: if the monoid 𝑀 has a 𝐶 logarithmic𝜃-power density with 0 < 𝜃 < 1, then the zeta function of the monoid 𝑀 has a holomorphic half-plane 𝜎 > 0 and the imaginary axis is the singularity
line.
In the third section, the question of the analytical continuation of the zeta function of the monoid of natural numbers in three cases is considered: for a monoid of 𝑘-th powers of natural numbers, for a set of natural numbers free of 𝑘-th powers, and for the union of two monoids of 𝑘-th powers of natural numbers when the exponents of the degrees are mutually prime numbers.
In all three cases, it is shown that the analytic continuation exists on the entire complex plane. Functional equations are found for each of the three cases. They all have a fundamentally different look. In addition, new properties of the zeta function that are missing from the Riemann zeta function are found for each analytic continuation in the critical band.
In conclusion, promising, relevant topics for further research are listed.