On monoids of natural numbers with unique factorization into prime elements

N. N. Dobrovolsky

Čebyševskij sbornik, Tome 19 (2018) no. 1, pp. 79-105

Résumé

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 $\beta>1$ examples of Dirichlet series with an abscissa of absolute convergence $\sigma=\frac{1}{\beta}$ are constructed. For any natural $\beta>1$ examples of a pair of zeta functions $\zeta(B|\alpha)$ and $\zeta(A_{B,\beta}|\alpha)$ with the equality $\sigma_{A_{B,\beta}}=\frac{\sigma_B}{\beta}$ 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.

Détail

Keywords: Riemann zeta function, Dirichlet series, zeta function of monoid of natural numbers, Euler product.