On monoids of natural numbers with unique factorization into prime elements
https://doi.org/10.22405/2226-8383-2018-19-1-79-105
Abstract
The paper continues research on a new class of Dirichlet series — zeta functions of monoids of natural numbers. The inverse Dirichlet series for zeta functions of monoids of natural numbers with unique factorization into prime elements and for zeta-functions of sets of prime elements of monoids with unique factorization into prime elements are studied.
For any β > 1 examples of Dirichlet series with an abscissa of absolute convergence σ = are constructed. For any natural β > 1 examples of a pair of zeta functions ζ(B|α) and ζ(AB,β|α) with the equality σAB,β = σB/ β are constructed.
Various examples of monoids and corresponding zeta functions of monoids are considered. A number of properties of the zeta functions of monoids of natural numbers with unique factorization into prime factors are obtained. An explicit form of the inverse series to the zeta-function of the set of primes supplemented by one is found.
An explicit form of the ratio of the Riemann zeta-function to the zeta-function of the set of primes supplemented by one is found.
Nested sequences of monoids generated by primes are considered. For the zeta-functions of these monoids the nesting principle is formulated, which allows to transfer the results about the coefficients of one zeta-functions to the coefficients of other zeta-functions.
In this paper the general form of all monoids of natural numbers with unique factorization into prime factors was described for the first time.
In conclusion, topical problems for zeta-functions of monoids of natural numbers that require further study are considered.
About the Author
N. N. Dobrovolsky
Tula State University
Russian Federation
Russian Federation
Dobrovolsky Nikolai Nikolaevich — candidate of physical and mathematical sciences, assistant of the department of applied mathematics and computer science
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Review
For citations:
Dobrovolsky N.N. On monoids of natural numbers with unique factorization into prime elements. Chebyshevskii Sbornik. 2018;19(1):79-105. (In Russ.) https://doi.org/10.22405/2226-8383-2018-19-1-79-105
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