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.

Bibliographie

[1] E. Bombieria, A. Ghoshb, “Around the Davenport-Heilbronn function”, Uspekhi Mat. Nauk, 66:2(398) (2011), 15–66 | MR

[2] S. M. Voronin, Izbrannye trudy: Matematika, ed. A. A. Karacuba, Izd-vo MGTU im. N. Je. Baumana, M., 2006, 480 pp.

[3] S. M. Voronin, A. A. Karacuba, Dzeta-funkcija Rimana, Izd-vo Fizmatlit, M., 1994, 376 pp. | MR

[4] M. N. Dobrovol'skij, “Funkcional'noe uravnenie dlja giperbolicheskoj dzeta-funkcii celochislennyh reshetok”, Doklady akademii nauk, 412:3 (2007), 302–304 | Zbl

[5] N. M. Dobrovolsky, N. N. Dobrovolsky, V. N. Soboleva, D. K. Sobolev, L. P. Dobrovol'skaya, O. E. Bocharova, “On hyperbolic Hurwitz zeta function”, Chebyshevskii Sbornik, 17:3 (2016), 72–105 | MR | Zbl

[6] N. N. Dobrovolsky, “The zeta-function is the monoid of natural numbers with unique factorization”, Chebyshevskii Sbornik, 18:4 (2017), 187–207 | MR

[7] N. N. Dobrovolsky, “On monoids of natural numbers with unique factorization into prime elements”, Chebyshevskii Sbornik, 19:1 (2018), 79–105 | MR

[8] N. N. Dobrovol'skii, “The zeta function of monoids with a given abscissa of absolute convergence”, Chebyshevskii sbornik, 19:3 (2018), 142–150 | MR | Zbl

[9] N. N. Dobrovol'skii, M. N. Dobrovol'skii, N. M. Dobrovol'skii, I. N. Balaba, I. Yu. Rebrova, “About «zagrobelna the series» for the zeta function of monoids with exponential sequence of simple”, Chebyshevskii sbornik, 19:3 (2018), 106–123 | MR | Zbl

[10] N. N. Dobrovol'skii, A. O. Kalinina, M. N. Dobrovol'skii, N. M. Dobrovol'skii, “On the number of prime elements in certain monoids of natural numbers”, Chebyshevskii sbornik, 2018, no. 2 | MR

[11] Prahar K., Raspredelenie prostyh chisel, per. s nem., Izd-vo Mir, M., 1967, 511 pp.

[12] I. Yu. Rebrova, A. V. Kirilina, “N. M. Korobov and the theory of the hyperbolic zeta function of lattices”, Chebyshevskii sbornik, 19:2 (2018) | MR

[13] C. Hooley, Nauka, M., 1987, 20 pp.

[14] P. L. Chebyshev, Complete works, v. I–V, Izd-vo AN SSSR, M.–L., 1944–1951 | MR

[15] P. L. Chebyshev, Selected works, Izd-vo AN SSSR, M., 1955, 926 pp. | MR

[16] H. Davenport, H. Heilbronn, “On the zeros of certain Dirichlet series”, J. London Math. Soc., 11 (1936), 181–185 | MR

[17] L. P. Dobrovolskaya, M. N. Dobrovolsky, N. M. Dobrovol'skii, N. N. Dobrovolsky, “On Hyperbolic Zeta Function of Lattices”, Continuous and Distributed Systems. Solid Mechanics and Its Applications, 211 (2014), 23–62 | DOI | MR | Zbl