Analytic Number Theory

In this paper, we argue an arithmetic issue also known as Ducci's 4 or nnumber game. First we set an arbitrary non-negative integer at each vertex of a polygon, and apply the following procedure: at the center of each edge, set the difference of two integers at both ends of the edge, and connect the centers of adjacent edges: in this way, we obtain a new polygon with non-negative integers at every vertex. We apply the same procedure recursively, until all numbers set around the polygon become zeros. In the case the number N of edges of the polygon is four, it has been claimed that the procedure will always terminate in finite steps, and it is proved by B. Freedman [1]. For an arbitrary N, we know the following facts. 1. If the number N of edges of the polygon is a power of two, the recursive procedure will always terminate in finite steps. 2. Otherwise, there exits some non-negative integers at vertices of the polygon, from which the recursive procedure will never terminate. This time, we discuss the following fact: Let A, B, or C be positive integers on consecutive vertices on polygon A≤C≤B, A≥C≥B, B≤A≤C, or B≥A≥C, the recursive procedure by the binary operation which vertices N are four will always terminate by four steps. In this paper, we prove it and also discuss further general case.

Bernoulli numbers which are ubiquitous in mathematics, typically appearing either as the Taylor coefficients of x/ tan x or else being very closed to this, as special values of the Riemann zeta function. But they also sometimes appear in other guises and in other combinations. Here we will look at the functional equation satisfied by the Bernoulli numbers. We begin with the less familiar and divergent ordinary generating series β(x) = ∞ n=0

This paper expounds the role of the non-trivial zeros of the Riemann zeta function ζ and supplements the author’s earlier papers on the Riemann hypothesis. There is a lot of mystery surrounding the non-trivial zeros. MSC: 11-XX (Number Theory)

People these days know the Universe as a Whole, because not knowing the edge between This and That. Its secret is the secret of a forgotten body, the seventh in the series of multifaceted as Life, Seven. We know six of it now. But the Ancestors knew all seven. // Люди ныне не знают Вселенной как Целого, не зная грани меж Этим и Тем. Тайна  ее есть тайна забытого тела, седьмого в строю многогранных как Жизнь, Семь. Мы знаем шесть ныне. А Пращуры знали все семь.

Number theory is a branch of mathematics that is primarily focused on the study of positive integers, or natural numbers, and their properties such as divisibility, prime factorization, or solvability of equations in integers. Number theory has a very long and diverse history, and some of the greatest mathematicians of all time, such as Euclid, Euler and Gauss, have made significant contributions to it. Throughout its long history, number theory has often been considered as the "purest" branch of mathematics in the sense that it was the furthest from any concrete application. However, a significant change took place in the mid-1970s, and nowadays, number theory is one of the most important branches of mathematics for applications in cryptography and secure information exchange. This book is based on teaching materials from the courses Number Theory and Elementary Number Theory, which are taught at the undergraduate level studies at the Department of Mathematics, Faculty of Science, University of Zagreb, and the courses Diophantine Equations and Diophantine Approximations and Applications, which were taught at the doctoral program of mathematics at that faculty. The book thoroughly covers the content of these courses, but it also contains other related topics such as elliptic curves, which are the subject of the last two chapters in the book. The book also provides an insight into subjects that were and are at the centre of research interest of the author of the book and other members of the Croatian group in number theory, gathered around the Seminar on Number Theory and Algebra. This book is primarily intended for students of mathematics and related faculties who attend courses in number theory and its applications. However, it can also be useful to advanced high school students who are preparing for mathematics competitions in which at all levels, from the school level to international competitions, number theory has a significant role, and for doctoral students and scientists in the fields of number theory, algebra and cryptography. In the English edition, there are only minor changes compared with the Croatian version. Several misprints noticed by the author and the readers were corrected. Some information and references were updated, in particular, those related to elliptic curves rank records and new constructions of families of rational Diophantine sextuples. At just a few places in the Croatian version of the book only the references to literature in Croatian were given ; these references were expanded in the English edition with the appropriate recommendations of literature in English. The list of references has been expanded to include some recent books and papers, as well as some references which were mentioned in the text of the Croatian edition but were not included in the list of references.

Motivated largely by a number of recent investigations, we introduce and investigate the various properties of a certain new family of the λ-generalized Hurwitz-Lerch zeta functions. We derive many potentially useful results involving these λ-generalized Hurwitz-Lerch zeta functions including (for example) their partial differential equations, new series and Mellin-Barnes type contour integral representations (which are associated with Fox's H-function) and several other summation formulas. We discuss their potential application in Number Theory by appropriately constructing a seemingly novel continuous analogue of Lippert's Hurwitz measure. We also consider some other statistical applications of the family of the λ-generalized Hurwitz-Lerch zeta functions in probability distribution theory.

Dirichlet's theorem states that there exist an infinite number of primes in an arithmetic progression a + mk when a and m are relatively prime and k runs over the positive integers. While a few special cases of Dirichlet's theorem, such as the arithmetic progression 2 + 3k, can be settled by elementary methods, the proof of the general case is much more involved. Analysis of the Riemann zeta-function and Dirichlet L-functions is used.

Ειδικό θέμα στις σειρές Dirichlet και στο Θεώρημα των Πρώτων αριθμών. Μελετούμε τις βασικές έννοιες των σειρών Dirichlet παρουσιάζοντας τα κύρια θεωρήματα , δίνουμε μία παραλλαγή της φόρμουλας του Perron, και κλείνουμε με την απόδειξη του Θεωρήματος των Πρώτων Αριθμών . Παραθέτουμε άλυτα προβλήματα στο τέλος της εργασίας.

In this paper we expand the prime number theorem, twice. Then we use both expansions to describe the distribution of primes. Afterwards we analyze the symmetry in the asymptotic equivalence. Before we come to the Weil conjecture, we derive a solution of the non-trivial zeros of the Riemann zeta function, the root identity, which we expand to the entire complex plane via a replication identity and which we employ to show, that the critical strip next to the critical line is a zero free zone.

Shapes defined by the golden ratio have long been considered aesthetically pleasing in western cultures, reflecting nature's balance between symmetry and asymmetry. The ratio is still used frequently in art and design. The golden ratio is also known as the golden mean, golden section, golden number or divine proportion.It is usually denoted by the Greek letter  (phi).

This paper consists of the extended working notes and observations made during the development of a joint paper [?] with Philippe Flajolet on the Riemann zeta function. Most of the core ideas of that paper, of which a majority are due to Flajolet, are reproduced here; however, the choice of wording used here, and all errors and omissions are my own fault. This set of notes contains considerably more content, although is looser and sloppier, and is an exploration of tangents, dead-ends, and ideas shooting off in uncertain directions.

This paper re-conceptualizes "information system success" as a formative, multidimensional index. Such a validated and widely accepted index would facilitate cumulative research on the impacts of IS, while at the same time provide a benchmark for organizations to track their IS performance. The proposed IS-Impact measurement model represents the stream of net benefits from an Information System (IS), to date and anticipated, as perceived by all key user groups. Model measures are formulated to be robust, economical, and simple, yielding results that are comparable across diverse systems and contexts, and from multiple user perspectives. The model includes four dimensions in two halves. The "impact" half measures benefits to date, or Individualand Organizational Impact; the "quality" half uses System Quality and Information Quality as proxies for probable future impacts. Study findings evidence the necessity, additivity, and completeness of these four dimensions.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ must always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. MSC: 11-XX (Number Theory)

3P derives from the three very important categories / areas in defining a reform process of the national security intelligence analysis, namely : – Process (the analysis activity, with his entire set of methods or means, internal procedures and standards, but also, with its variou s types of organization); – Product (the results of the analysis activity, the p roducts which are sent to beneficiaries / users and the feedback or the requests for inform ation from the intelligence consumers); – Personnel (the intelligence analyst, as well as the process of its selection and training). Why these three P? I must confess that the idea of trying to define the analysis activity, and implicitly, the main parts of a reform in this important area for the activity of each intelligence service came following some discussion I had with colleagues, experts and intelligence analysts. I worked, as co-author, to a paper (presented within an international forum) concerning the evolution of intelligenc...

This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's "An efficient algorithm for computing the Riemann zeta function" 5], to more general series. The algorithm provides a rapid means of evaluating Li s (z) for general values of complex s and a kidney-shaped region of complex z values given by z 2 /(z − 1) < 4. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values.

This paper, which is published in an international mathematics journal and endorsed by a mathematics society, touches on the part played by the non-trivial zeros of the Riemann zeta function ζ, providing many important information and insights in the process, including some approaches to the Riemann hypothesis.

In this article we describe some elements from the chapter of Mathematics " Number Theory " , as they appear in the Codex Vindobonensis phil. Gr. 65, a Byzantine Ms kept in the National Library of Austria in Vienna. This codex contains a comprehensive program of teaching Mathematics which was addressed to an audience consisting of students probably of all the grades of what is today's primary and secondary education, but also state functionaries, merchants, craftsmen of various specialities such as silversmiths and goldsmiths, builders, etc. The mathematical branch of Number Theory and the numerical position system is integrated in codex 65 in the respective chapters of the four operations and their checks. Introduction At the beginning of this article we briefly describe Codex Vindobonensis phil. Gr. 65 fols 11r ―126r, which was written circa AD 1436 by an anonymous author, and was intended for teaching purposes. We record the syllabus units so that the reader will form an initial idea about the content of the Ms, and then we analyze in detail the problems related to the Number Theory, but also more generally to the numerical system of position in Byzantium.

Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri-Lagarias, Maślanka, Coffey, Báez-Duarte, Voros and others. We apply the theory of Nörlund-Rice integrals in conjunction with the saddlepoint method and derive precise asymptotic estimates. The method extends to Dirichlet Lfunctions and our estimates appear to be partly related to earlier investigations surrounding Li's criterion for the Riemann hypothesis. √ πn .

This paper shows why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. [The paper is published in a journal of number theory.]

Extending a previous result, we show that, for the friable summation method, the Fourier series of any normalized function F with bounded variation on the unidimensional torus converges pointwise to F while avoiding the Gibbs phenomenon. We also prove that the convergence is uniform when F is continuous and provide an e↵ective bound for the speed when F satisfies a uniform Lipschitz condition.

The objective of this study is to explore an innovative pattern along with derivations connected to the Fibonacci sequences. Relevant proofs with supporting explanations have been provided in this study to derive the conceptualized pattern for this study which in turn will ensure reduction of the complexions and simplification of the Fibonacci formula while solving different problems linked to Fibonacci series and Fibonacci-type sequences. An attempt has also been made in this study to validate the conceptualized pattern with the help of induction method.

In this paper, we find all the solutions of the Diophantine equation P +P m +P n = 2 a , in nonnegative integer variables (n, m, , a) where P k is the k-th term of the Pell sequence {P n } n≥0 given by P 0 = 0, P 1 = 1 and P n+1 = 2P n +P n−1 for all n ≥ 1. MSC: 11D45, 11B39; 11A25

The paper discusses some advances on prime numbers. On the first observation we learn to span all odd composite numbers on the number line in terms of x and y both ∈ N. This gives us an algebraic form that holds only a prime at the point of no solution and thus, helped us span the primes in negation. In the process we could address specifically those numbers that are separated from an odd composite by a gap k in terms of a union of congruence relations. The number of primes in this set is the difference between the number of primes considered let's say N and the number of the prime gap k occurring till the Nth prime. This union can be further reduced to a series of intersections each of which has a unique solution from the Chinese remainder theorem. This then connects to the prime counting functions for a line to the frequency of the prime gap. We show a relation between the prime gaps of 2 and 4 occurring equally often in N and the prime number theorem in arithmetic progressions.

Let b ≥ 2 be a given integer. In this paper, we show that there only finitely many positive integers d which are not squares, such that the Pell equation X 2 −dY 2 = 1 has two positive integer solutions (X, Y) with the property that their X-coordinates are base b-repdigits. We also give an upper bound on the largest such d in terms of b.