On the monoid of quadratic residues

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

Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 95-108

Résumé

In this paper we study the Zeta function of the monoid of quadratic residues modulo a simple $p$. The monoid of quadratic residues is given by $$ M_{p, 2}=\left\{a\in\mathbb{N}\left| \left(\frac{a}{p}\right)=1\right.\right\}=\bigcup_{\nu=1}^{\frac{p-1}{2}}\left (r_\nu+p\mathbb{N}_0\right), $$ where $\mathbb{N}_0=\{0\}\bigcup\mathbb{N}$ and $r_1$ — the smallest positive system of quadratic residues modulo $p$, respectively, $r_{\frac{p+1}{2}}\ldots$ — the smallest positive system of quadratic residuals modulo $p$. The set of simple elements of a monoid $M_{p, 2}$ consists of a set of Prime numbers $\mathbb{P}_p^{(1)}$ and a set of pseudo-Prime numbers $\mathbb{P}_p^{(2)} \cdot\mathbb{P}_p^{(2)}$: $$ P (M_{p,2})=\mathbb{P}_p^{(1)}\bigcup\left(\mathbb{P}_p^{(2)}\cdot\mathbb{P}_p^{(2)}\right), $$ where the Prime set $\mathbb{P}$ is split into two infinite subsets $\mathbb{P}_p^{(\nu)}$ $(\nu=1,2)$ and the singleton set $\{p\}$: $$ \mathbb{P}=\mathbb{P}_p^{(1)}\bigcup\mathbb{P}_p^{(2)}\bigcup\{p\}, \quad \mathbb{P}_p^{(\nu)}=\left\{q\in\mathbb{P}\left|\left(\frac{q}{p}\right)=3-2\nu\right.\right\} \quad (\nu=1,2). $$ The monoid $M_{p, 2}$ decomposes into a product of two mutually simple monoids $M_{p, 2}=M_{p,2}^{(1)}\cdot M_{p,2}^{(2)}$, where $$ M_{p, 2}^{(\nu)}=\left\{a\in M_{p,2}\left| a=\prod_{j=1}^{n}q_j^{\alpha_j}, \, q_j\in\mathbb{P}_p^{(\nu)} \right.\right\}, \quad \nu=1,2. $$ The paper studies the properties of the distribution function of simple elements $\pi_{M_{p, 2}^{(\nu)}} (x)$ for $\nu=1,2$. Note that $\pi_{M_{p, 2}} (x)=\pi_{M_{p,2}^{(1)}}(x)+\pi_{M_{p,2}^{(2)}}((x)$. It is shown that $$ \pi_{M_{p,2}^{(1)}}(x)=\frac{1}{2}\mathrm{li} x+O\left(\frac{x^{\beta_1}}{2}+\frac{p-1}2xe^{-c_9\sqrt{\ln x}}\right) $$ and $$ \pi_{M_{p,2}^{(2)}}(x)=\frac{x\ln\ln x}{2\ln x}+O\left(\frac{x}{(1 - \beta_1)\ln{x}}\right), $$ where $\beta_1$ — exceptional zero of exceptional character $\chi_1$ modulo $p$. In conclusion, the actual problems with Zeta functions of monoids of natural numbers requiring further research are considered.

Détail

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