Hypothesis about "barier series" for the zeta-functions of monoids with the exponential sequence of primes

N. N. Dobrovolsky ; M. N. Dobrovolsky ; N. M. Dobrovolsky ; I. N. Balaba ; I. Yu. Rebrova

Čebyševskij sbornik, Tome 19 (2018) no. 1, pp. 106-123

Résumé

The work continues the study of a new class of Dirichlet series — the zeta functions of monoids of natural numbers. First of all, we study in detail the zeta function $\zeta(M(q)|\alpha)$ of geometric progression $M(q)$ with initial value equal to 1 and an arbitrary natural common ratio $q>1$, which is the simplest monoid of natural numbers with a unique decomposition into prime elements of the monoid. For a meromorphic function $\zeta(M(q)|\alpha)=\frac{q^\alpha}{q^\alpha-1}$, which have poles $$ S(M (q))=\left\{\left. \frac{2\pi i k}{\ln q}\right| k\in\mathbb{Z}\right\} $$ representations are received: \begin{gather*} \zeta(M(q)|\alpha)=\frac{q^{\frac{\alpha}{2}}}{\alpha\ln q}\prod_{n=1}^{\infty}\left(1+\frac{\alpha^2\ln^2 q}{4\pi^2 n^2}\right)^{-1}=\frac{1}{2}+\frac{1}{\alpha\ln q}+\sum_{n=1}^{\infty}\frac{2\alpha\ln q}{\alpha^2\ln^2 q+4n^2\pi^2}= \\ =\frac{q^{\frac{\alpha}{2}}\alpha\ln q}{4\pi^2}\Gamma\left(\frac{\alpha i\ln q }{2\pi}\right)\Gamma\left(-\frac{\alpha i\ln q }{2\pi}\right). \end{gather*} For the zeta function $\zeta(M(\vec{p})|\alpha)$ of the monoid $M(\vec{p})$ with a finite number of primes $\vec{p}=(p_1,\ldots,p_n)$ the decomposition into an infinite product is obtained $$ \zeta(M(\vec{p})|\alpha)=\frac{P(\vec{p})^{\frac{\alpha}{2}}}{\alpha^nQ(\vec{p})}\prod_{\nu=1}^{n}\prod_{m=1}^{\infty}\left(1+\frac{\alpha^2\ln^2 p_\nu}{4\pi^2 m^2}\right)^{-1}, $$ where $P (\vec{p})=p_1\ldots p_n$, $Q (\vec{p})=\ln p_1\ldots \ln p_n$, and a functional equation is found $$ \zeta (M (\vec{p})|-\alpha)=(-1)^n\frac{\zeta (M (\vec{p})|\alpha)}{P (\vec{p})^\alpha}. $$ For the monoid of natural numbers $M^*(\vec{p})= \mathbb{N}\cdot M^{-1} (\vec{p})$ with a unique decomposition into prime elements, which consists of natural numbers $n$ coprime with $P (\vec{p})=p_1\ldots p_n$, and for the Euler product $P (M^*(\vec{p}) / \ alpha)$, which consists of factors for all primes other than $p_1,\ldots, p_n$, a functional equation is found $$ \zeta(M^*(\vec{p})|\alpha)=M(\vec{p},\alpha) \zeta(M^*(\vec{p})|1-\alpha), $$ where $$ M(\vec{p},\alpha)=M(\alpha)\cdot\frac{M_1(\vec{p},\alpha)}{M_1(\vec{p},1-\alpha)}, \quad M_1(\vec{p},\alpha)=\prod_{\nu=1}^{n}\left(1-\frac{1}{p_\nu^\alpha}\right). $$ It is proved that for any infinite set of primes $\mathbb{P}_1$ there is no analytic function equal to $$\lim\limits_{n\to\infty} \zeta(M(\vec{p}_n)|\alpha)$$ on the whole complex plane. The hypothesis about the barrier series for any exponential set of $ PE $ prime numbers is formulated. 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 monoids of natural numbers, Euler product, logarithm of the Euler product.