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

This article is cited in 4 scientific papers (total in 4 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
$$ M_{p, 2}=\{a\in\mathbb{N}| (\frac{a}{p})=1.\}=\bigcup_{\nu=1}^{\frac{p-1}{2}}(r_\nu+p\mathbb{N}_0), $$
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 $\mathbb{P}_p^{(2)} \cdot\mathbb{P}_p^{(2)}$:
$$ P (M_{p,2})=\mathbb{P}_p^{(1)}\bigcup(\mathbb{P}_p^{(2)}\cdot\mathbb{P}_p^{(2)}), $$
where the Prime set $\mathbb{P}$ is split into two infinite subsets $\mathbb{P}_p^{(\nu)}$ $(\nu=1,2)$ and the singleton set $\{p\}$:
$$ \mathbb{P}=\mathbb{P}_p^{(1)}\bigcup\mathbb{P}_p^{(2)}\bigcup\{p\}, \quad \mathbb{P}_p^{(\nu)}=\{q\in\mathbb{P}|(\frac{q}{p})=3-2\nu.\} \quad (\nu=1,2). $$
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
$$ M_{p, 2}^{(\nu)}=\{a\in M_{p,2}| a=\prod_{j=1}^{n}q_j^{\alpha_j},   q_j\in\mathbb{P}_p^{(\nu)} .\}, \quad \nu=1,2. $$
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
$$ \pi_{M_{p,2}^{(1)}}(x)=\frac{1}{2}\mathrm{li} x+O(\frac{x^{\beta_1}}{2}+\frac{p-1}2xe^{-c_9\sqrt{\ln x}}) $$
and
$$ \pi_{M_{p,2}^{(2)}}(x)=\frac{x\ln\ln x}{2\ln x}+O(\frac{x}{(1 - \beta_1)\ln{x}}), $$
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


DOI: https://doi.org/10.22405/2226-8383-2018-19-3-95-108

Full text: PDF file (745 kB)
References: PDF file   HTML file

UDC: 511.3
Received: 30.06.2018
Accepted:15.10.2018

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 articles:
    1. N. N. Dobrovolskii, “Odna modelnaya dzeta-funktsiya monoida naturalnykh chisel”, Chebyshevskii sb., 20:1 (2019), 148–163  mathnet  crossref
    2. N. N. Dobrovolskii, M. N. Dobrovolskii, N. M. Dobrovolskii, I. N. Balaba, I. Yu. Rebrova, “Algebra ryadov Dirikhle monoida naturalnykh chisel”, Chebyshevskii sb., 20:1 (2019), 180–196  mathnet  crossref
    3. N. N. Dobrovolskii, N. M. Dobrovolskii, I. Yu. Rebrova, A. V. Rodionov, “Monoidy naturalnykh chisel v teoretiko-chislovom metode v priblizhennom analize”, Chebyshevskii sb., 20:1 (2019), 164–179  mathnet  crossref
    4. N. N. Dobrovolskii, M. N. Dobrovolskii, N. M. Dobrovolskii, “Ob odnom obobschennom eilerovom proizvedenii, zadayuschem meromorfnuyu funktsiyu na vsei kompleksnoi ploskosti”, Chebyshevskii sb., 20:2 (2019), 156–168  mathnet  crossref
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