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Chebyshevskii Sbornik, 2018, Volume 19, Issue 3, Pages 95–108
DOI: https://doi.org/10.22405/2226-8383-2018-19-3-95-108
(Mi cheb682)
 

This article is cited in 11 scientific papers (total 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
References:
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.
Funding agency Grant number
Russian Foundation for Basic Research 16-41-710194_р_центр_а
Received: 30.06.2018
Accepted: 15.10.2018
Bibliographic databases:
Document Type: Article
UDC: 511.3
Language: Russian
Citation: N. N. Dobrovolsky, A. O. Kalinina, M. N. Dobrovolsky, N. M. Dobrovolsky, “On the monoid of quadratic residues”, Chebyshevskii Sb., 19:3 (2018), 95–108
Citation in format AMSBIB
\Bibitem{DobKalDob18}
\by N.~N.~Dobrovolsky, A.~O.~Kalinina, M.~N.~Dobrovolsky, N.~M.~Dobrovolsky
\paper On the monoid of quadratic residues
\jour Chebyshevskii Sb.
\yr 2018
\vol 19
\issue 3
\pages 95--108
\mathnet{http://mi.mathnet.ru/cheb682}
\crossref{https://doi.org/10.22405/2226-8383-2018-19-3-95-108}
\elib{https://elibrary.ru/item.asp?id=39454391}
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  • This publication is cited in the following 11 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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