RUS  ENG
Full version
JOURNALS // Chebyshevskii Sbornik // Archive

Chebyshevskii Sb., 2018 Volume 19, Issue 3, Pages 95–108 (Mi cheb682)

This article is cited in 11 papers

On the monoid of quadratic residues

N. N. Dobrovolskyab, A. O. Kalininac, M. N. Dobrovolskyd, N. M. Dobrovolskyb

a Tula State University
b Tula State Pedagogical University
c Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
d Geophysical center of RAS

Abstract: 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 $r_1<r_2<\ldots<r_ {\frac{p-1}{2}}$ — the smallest positive system of quadratic residues modulo $p$, respectively, $r_{\frac{p+1}{2}}<\ldots<r_{p-1}$ — 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.

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

UDC: 511.3

Received: 30.06.2018
Accepted: 15.10.2018

DOI: 10.22405/2226-8383-2018-19-3-95-108



Bibliographic databases:


© Steklov Math. Inst. of RAS, 2025