On the number of prime elements in certain monoids of natural numbers

N. N. Dobrovolsky ; A. O. Kalinina ; M. N. Dobrovolsky ; N. M. Dobrovolsky

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 123-141

Résumé

In this paper we study the number of prime elements in the monoid $M_{q, 1}$ consisting of natural numbers comparable to $1$ modulo $q$. For $q>2$, the monoid $M_{q, 1}$ is not a monoid with a unique decomposition into prime elements, since along with ordinary primes that are comparable to $1$ modulo $q$, pseudo-primes that are composite numbers fall into the number of prime elements. The case $q=3,4,6$ is distinguished from the others by the fact that pseudo-primes are the product of two primes comparable to $q-1$ modulo $q$. Thus, in this case for the set of prime elements $P(M_{q,1})$ of monoid $M_{q,1}$ the equality $P(M_{q,1})=\mathbb{P}_{q,1}\bigcup(\mathbb{P}_{q,q-1}\cdot\mathbb{P}_{q,q-1})$ is true. Since the monoid $M_{q,1}$ does not have the uniqueness of decomposition into prime elements, then the Zeta-function $$ \zeta (M_{q,1}|\alpha)=\sum_{n\in M_{q, 1}} \frac{1}{n^\alpha} $$ of the monoid $M_{q, 1}$ is not equal to the Euler product $$ P(M_{q,1}|\alpha)=\prod_{r\in P(M_{q,1})}\left (1-\frac{1}{r^\alpha}\right)^{-1}. $$ Therefore, it is not possible to study the distribution of prime elements in the monoid $M_{q, 1}$ using the analytical properties of the logarithmic derivative of the zeta function of the monoid. For completeness, the paper first studies the question of the number of composite numbers equal to the product of two primes using Chebyshev's inequalities, since this year marks the 170th anniversary of the release of the first memoir of P. L. Chebyshev about primes. Then, using the Brun-Titchmarsh inequality, we obtain an upper bound on the number of composite numbers comparable to $1$ modulo $q$ and equal to the product of two primes. The approach applied to the general case is then transferred to the case of prime elements in monoids $M_{q, 1}$ with $q=3,4,6$. In conclusion, topical problems with zeta-functions of monoids of natural numbers that require further investigation are considered.

Détail

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