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Symmetry
Special Issues
Number Theory and Discrete Mathematics
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Number Theory and Discrete Mathematics

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 November 2021) | Viewed by 6480

Special Issue Editors

Prof. Elena Deza

E-Mail Website
Guest Editor
Department of Mathematics, Moscow State Pedagogical University, Moscow, Russia
Interests: analytic number theory (dirichlet divisor problem, dirichlet series, theory of Riemann's zeta function, mean values of arithmetical functions); discrete mathematics (special integer numbers, combinatorics, graph theory); theory of discrete metric spaces (oriented metrics, m-metrics, partial metrics); problems of mathematical education
Prof. Dr. Dobrovol’skii Nikolai Mihailovich

E-Mail Website
Guest Editor
Tula State L. N. Tolstoy Pedagogical University, Tula, Russia
Interests: algebra, mathematical analysis and geometry

Special Issue Information

Dear Colleagues,

It is well known that modern Mathematics naturally combines classical and new fields. Number Theory certainly represents the classical Mathematics. Discrete Analysis is a part of the modern Mathematics. At the same time, we can obtain very interesting results by studying problems at the border of Number Theory and Discrete Mathematics. Consider, for example, the Theory of Special Numbers. On the one hand, the numerical nature of these objects indicates their belonging to the Number Theory. On the other hand, the structure of these sets, the methods of their research, the problems in which they arise, often have a discrete character. Thus, the study of Stirling numbers, Catalan numbers, Bernoulli numbers etc. contributes to both classical Number Theory and Discrete Mathematics. For our special issue, we are looking for articles on classical problems of Number Theory and modern problems of Discrete Analysis. But the most interesting for us are articles containing solutions of problems combining these two areas of Mathematical Science.

Prof. Elena Deza
Prof. Dobrovol’skii Nikolai Mihailovich
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Analytic Number Theory
  • Theory of prime numbers
  • Theory of zeta-functions
  • Sieve theory
  • Algebraic number theory
  • Algebraic number fields
  • Diophantine equations
  • Diophantine geometry
  • Additive number theory
  • Cryptography
  • Combinatorics
  • Discrete probability Theory
  • Arithmetic Combinatorics
  • Graph theory
  • Discrete geometry

Published Papers (3 papers)

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Research

14 pages, 262 KiB  
Article
The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function
by Janyarak Tongsomporn, Saeree Wananiyakul and Jörn Steuding
Symmetry 2021, 13(12), 2410; https://doi.org/10.3390/sym13122410 - 13 Dec 2021
Cited by 4 | Viewed by 1770
Abstract
In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is [...] Read more.
In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally. Full article
(This article belongs to the Special Issue Number Theory and Discrete Mathematics)
11 pages, 293 KiB  
Article
On Independent Secondary Dominating Sets in Generalized Graph Products
by Adrian Michalski and Paweł Bednarz
Symmetry 2021, 13(12), 2399; https://doi.org/10.3390/sym13122399 - 12 Dec 2021
Cited by 3 | Viewed by 1871
Abstract
In 2008, Hedetniemi et al. introduced (1,k)-domination in graphs. The research on this concept was extended to the problem of existence of independent (1,k)-dominating sets, which is an NP-complete problem. In this [...] Read more.
In 2008, Hedetniemi et al. introduced (1,k)-domination in graphs. The research on this concept was extended to the problem of existence of independent (1,k)-dominating sets, which is an NP-complete problem. In this paper, we consider independent (1,1)- and (1,2)-dominating sets, which we name as (1,1)-kernels and (1,2)-kernels, respectively. We obtain a complete characterization of generalized corona of graphs and G-join of graphs, which have such kernels. Moreover, we determine some graph parameters related to these sets, such as the number and the cardinality. In general, graph products considered in this paper have an asymmetric structure, contrary to other many well-known graph products (Cartesian, tensor, strong). Full article
(This article belongs to the Special Issue Number Theory and Discrete Mathematics)
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13 pages, 375 KiB  
Article
Riemann Hypothesis and Random Walks: The Zeta Case
by André LeClair
Symmetry 2021, 13(11), 2014; https://doi.org/10.3390/sym13112014 - 23 Oct 2021
Cited by 4 | Viewed by 1908
Abstract
In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its L-function is valid to the right of the critical line [...] Read more.
In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its L-function is valid to the right of the critical line ℜ(s)>12, and the Riemann hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line. Full article
(This article belongs to the Special Issue Number Theory and Discrete Mathematics)
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