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(A B, β |α ) with the equality σ AB,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.
Highlights
A number of properties of the zeta functions of monoids of natural numbers with unique factorization into prime factors are obtained
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
Summary
В данной работе продолжаются исследования из работы [8] и сохраняются обозначения и определения из этой работы. Будем через P3,1 и P3,2 обозначать множество всех простых чисел вида 3n + 1 и 3n + 2, соответствено. В частности, в это множество псевдопростых чисел входят квадраты простых. Множество простых элементов P (M3,1,2,0) состоит из псевдопростых чисел вида p1p2, где p1, p2 — произвольные различные простые числа вида 3m + 2. Тогда для произвольного моноида M натуральных чисел с однозначным разложением на простые элементы справедливо равенство ζ(M |α) = P (M |α). Через k(x) будем обозначать количество различных канонических представлений числа x, тогда эйлерово произведение P (M |α) будет раскладываться в следующий ряд Дирихле. Равенство эйлерова произведения и дзета-функции моноида M равносильно однозначности разложения на простые элементы в этом моноиде. Цель данной статьи — описать моноиды натуральных чисел с однозначным разложением на простые элементы, изучить их свойства, дзета-функции этих моноидов натуральных чисел и найти их обратные ряды Дирихле
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