DOCSLIB.ORG

A Weighted Version of the Erdős-Kac Theorem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Weighted Version of the Erdős-Kac Theorem A WEIGHTED VERSION OF THE ERDOS-KAC} THEOREM RIZWANUR KHAN, MICAH B. MILINOVICH, AND UNIQUE SUBEDI Abstract. Let !(n) denote the number of distinct prime factors of n and let τk(n) denote the k-fold divisor function. We evaluate the centralized moments of !(n), weighted by τk(n), and deduce a weighted version of the Erd}os-KacTheorem. 1. Introduction Let !(n) denote the number of distinct prime factors of natural number n. That is, X !(n) = 1: pjn The celebrated Erd}os{Kactheorem [6] from 1940 states that for each α 2 R, we have Z α 1 X 1 2 (1.1) 1 −! p e−t =2 dt x 2π −∞ n≤x p !(n)−log log x≤α log log x as x ! 1. In other words, since the log log function grows very slowly, the quantity !(n) − log log n (1.2) p log log n follows a Gaussian distribution law with mean 0 and variancep 1. R´enyi and Tur´an[17] provided a (best possible) quantitative version of the theorem with O(1= log log x) as the rate of convergence in (1.1). Although the Erd}os{Kac theorem is a classical result, it has remained a source of great research interest even in modern times. Many notable mathematicians have revisited the theorem and pro- vided different proofs of it, e.g. [6, 14, 17, 10, 1, 9, 13, 11]. We highlight a couple of these approaches in particular. Generalizing the prime counting function π(x), let πk(x) denote the number of integers n ≤ x with !(n) = k. One approach to the Erd}os{Kactheorem, related to the proof of R´enyi and Tur´an[17] and thep Selberg{Delange Method [16, Chapter 7.4], is to use asymptotics for πk(x) with k ≤ log log x + α log log x to evaluate the left hand side of (1.1). Such a proof is rather involved, requiring complex analysis and the theory of the Riemann zeta function, and it is at least as deep as the prime number theorem. Another approach is to asymptotically evaluate the moments of the quantity (1.2) and show that they match the moments of a standard Gaussian random variable. Since a Gaussian distribution is completely characterized by its moments [2, Theorem 30.1], this implies the Erd}os{KacTheorem. The moments approach was first accomplished by Halberstam [10] in 1955, although his proof was rather complicated. About a decade later, Billingsley [1] gave a much simpler demonstration. In 2007, Granville and Soundararajan [9] provided an even more transparent and flexible treatment. Their approach remains one of the most direct and elementary ways to prove the Erd}os-Kactheorem. 2020 Mathematics Subject Classification. 11N60. Key words and phrases. Erd}os{Kactheorem, prime omega function, divisor function, central limit theorem, moments. 1 2 RIZWANUR KHAN, MICAH B. MILINOVICH, AND UNIQUE SUBEDI This paper is concerned with a generalization of the Erd}os-Kactheorem in which the distribution of !(n) is studied, but counting is weighted by the divisor function. Elliott [4, 5] proved that1 −1 Z α X X 1 2 (1.3) τ(n) τ(n) −! p e−t =2 dt; 2π −∞ n≤x n≤x p !(n)−2 log log x≤α 2 log log x P where τ(n) = djn 1 is the divisor function. Thus, when weighted by the divisor function, !(n) still follows a Gaussian distribution as in the Erd}os{Kactheorem, but with double the mean and double the variance. Such a result can be predicted by the following heuristic. Recall that τ(n) = 2!(n) for square-free n. So, roughly speaking, we are studying the Gaussian distributed !(n), tilted by its exponential. Consider a Gaussian random variable with mean 0 and variance 1, so that its distribution function is α 1 Z −x2 p e 2 dx: 2π −∞ If we weight the measure by ex, the distribution function becomes p Z α 2 Z α 2 1 −x x e −(x−1) p e 2 e dx = p e 2 dx; 2π −∞ 2π −∞ where equality is obtained by completing the square. Thus, the resulting distribution with the weighted measure is still Gaussian but with altered mean and variance. In Probability theory, this phenomenon is a simple case of Girsanov's Theorem [12, Chapter 3.5]. We also mention that Elliott's P result in (1.3) was generalized to short interval sums x≤n≤x+y, where y is a small power of x, by Liu and Wu [15], using similar methods. Weighted central limit theorems are also of current interest in other parts of number theory. For 1 instance, there is a famous and classical result of Selberg which establishes that log jζ( 2 + it)j has a Gaussian distribution for t 2 [T; 2T ] as T ! 1, where ζ(s) denotes the Riemann zeta-function. Recently, Fazzari [7, 8] has proved weighted versions of Selberg's central limit theorem assuming the Riemann hypothesis. Our paper, which was mostly written before we were aware of Elliot's previous work, was inspired by Fazzari's results for the Riemann zeta-function. Elliott's proof of (1.3) is essentially based on a Selberg{Delange type approach to the Erd}os{Kac theorem, and thus is quite deep. The goal of this paper is to give a relatively simple proof of Elliott's result using a moments approach. Having a moments result to this weighted problem is natural, given the historical development of the Erd}os{Kactheorem. This was lacking in the literature prior to our work. Also, while Elliott's weighting was by the divisor function, we generalize this result to weighting by the k-fold divisor function, τk(n). We compute the centralized moments of !(n) weighted by τk(n) by generalizing a method of Granville and Soundararajan [9]. Although the foundation of our approach was laid out elegantly in [9], our proof requires a number of nontrivial modifications. It was not clear, a priori, whether the Granville-Soundararajan approach would work in this case and our generalization requires a careful set-up. We now state our main theorem. Let X τk(n) = 1 n1···nk=n be the k-fold divisor function. In Section 2.2, we show that, as n ranges over the integers below x, the mean of !(n) with respect to the weighted measure τk(n) is asymptotic to k log log x. Thus we centralize by k log log x and evaluate the following m-th moment for (!(n) − k log log x), weighted by τk(n). 1Elliott more generally proved (1.3) with τ(n)α for any α > 0. A WEIGHTED VERSION OF THE ERDOS-KAC} THEOREM 3 Theorem 1.1. Let k and m be fixed natural numbers. For x > 3, we have X m !(n) − k log log x τk(n) n≤x X τk(n) (1.4) n≤x ( m=2 m−1 (m − 1)!! k log log x + O (log log x) 2 ; if m is even; = m−1 O (log log x) 2 ; if m is odd; where (m − 1)!! denotes the product of all odd integers up to and including (m − 1). Dividing both sides by (k log log x)m=2, this gives that the weighted m-th moment of !(n) − k log log x p k log log x is (m − 1)!! + o(1) if m is even and is o(1) if m is odd. Recall that the m-th moment of a standard Gaussian random varriable is (m − 1)!! if m is even and is 0 if m is odd. Thus, Theorem 1.1 implies the following weighted version of the Erd}os{KacTheorem. Corollary 1.2. Fix k; m 2 N. For any α 2 R; we have −1 Z α X X 1 −t2=2 (1.5) τk(n) τk(n) −! p e dt 2π −∞ n≤x n≤x p !(n)−k log log x≤α k log log x as x ! 1. Throughout this paper, we will follow the "-convention, where " will always denote an arbitrarily small positive constant, but not necessarily the same one from one occurrence to the next. Our error terms are allowed to depend on ". 2. Preliminaries 2.1. Averages of the k-fold divisor function. A well known result, due to Voronoi and Landau (see [19, Theorem 12.2]), states that for s X x k k−1 +" (2.1) τk(n) = Res ζ (s) + O x k+1 s=1 s n≤x for x ≥ 1 and k ≥ 2. The leading order term of the residue is x (log x)k−1 ; (k − 1)! while the lower order terms are proportional to x(log x)k−1−c for integers c with 1 ≤ c ≤ k − 1. We will need to sum τk(n) with n divisible by a fixed a 2 N. We prove such a result with a weaker error term than (2.1). This suffices for our application, as we only require a power savings in x=a below. Lemma 2.1. For a 2 N and x ≥ a, we have s k+3 X x k x k+6 +" !(a) (2.2) τk(n) = Res ζ (s)F (s; a) + O τk(a) M ; s=1 s a n≤x ajn where υ −1 k p m Y 1 X τk(p ) (2.3) F (s; a) := 1 − 1 − ps pms pυp jja m=0 4 RIZWANUR KHAN, MICAH B. MILINOVICH, AND UNIQUE SUBEDI and p 2 + 16 (2.4) M := p : 2 − 1 Though the function F (s; a) depends on k, we suppress this in the notation since k is assumed to be fixed.
Load more

Recommended publications
  • Counting Restricted Partitions of Integers Into Fractions: Symmetry and Modes of the Generating Function and a Connection to Ω(T)
    Counting Restricted Partitions of Integers into Fractions: Symmetry and Modes of the Generating Function and a Connection to !(t) Z. Hoelscher and E. Palsson ∗ December 1, 2020 Abstract - Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions for integers smaller than the denominator form symmetric patterns. If the number of terms is restricted to h, then the nonzero terms of the generating function are unimodal, with the integer h having the most partitions. Such properties can be applied to a particular class of nonlinear Diophantine equations. We also examine partitions with even numerators. We prove that there are 2!(t) − 2 partitions of an integer t into fractions with the first x consecutive even integers for numerators and equal denominators of y, where 0 < y < x < t. We then use this to produce corollaries such as a Dirichlet series identity and an extension of the prime omega function to the complex plane, though this extension is not analytic everywhere. Keywords : Integer partitions; Restricted partitions; Partitions into fractions Mathematics Subject Classification (2020) : 05A17 1 Introduction Integer partitions are a classic part of number thoery. In the most general, unrestricted case, one seeks to express positive integers as the sum of smaller positive integers. Often the function p(n) is used to denote the count of the partitions of n. No simple formula for this is known, though a generating function can be written [1].
    [Show full text]
  • Generalizing Ruth-Aaron Numbers
    Generalizing Ruth-Aaron Numbers Y. Jiang and S.J. Miller Abstract - Let f(n) be the sum of the prime divisors of n, counted with multiplicity; thus f(2020) = f(22 ·5·101) = 110. Ruth-Aaron numbers, or integers n with f(n) = f(n+1), have been an interest of many number theorists since the famous 1974 baseball game gave them the elegant name after two baseball stars. Many of their properties were first discussed by Erd}os and Pomerance in 1978. In this paper, we generalize their results in two directions: by raising prime factors to a power and allowing a small difference between f(n) and f(n + 1). We x(log log x)3 log log log x prove that the number of integers up to x with fr(n) = fr(n+1) is O (log x)2 , where fr(n) is the Ruth-Aaron function replacing each prime factor with its r−th power. We also prove that the density of n remains 0 if jfr(n) − fr(n + 1)j ≤ k(x), where k(x) is a function of x with relatively low rate of growth. Moreover, we further the discussion of the infinitude of Ruth-Aaron numbers and provide a few possible directions for future study. Keywords : Ruth-Aaron numbers; largest prime factors; multiplicative functions; rate of growth Mathematics Subject Classification (2020) : 11N05; 11N56; 11P32 1 Introduction On April 8, 1974, Henry Aarony hit his 715th major league home run, sending him past Babe Ruth, who had a 714, on the baseball's all-time list.
    [Show full text]
  • Mathematical Constants and Sequences
    Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) ..
    [Show full text]
  • Series of Reciprocal Powers of K-Almost Primes
    SERIES OF RECIPROCAL POWERS OF k−ALMOST PRIMES RICHARD J. MATHAR Abstract. Sums over inverse s-th powers of semiprimes and k-almost primes are transformed into sums over products of powers of ordinary prime zeta functions. Multinomial coefficients known from the cycle decomposition of permutation groups play the role of expansion coefficients. Founded on a known convergence acceleration for the ordinary prime zeta functions, the sums and first derivatives are tabulated with high precision for indices k = 2,..., 6 and integer powers s = 2,..., 8. 1. Overview Series over rational polynomials evaluated at integer arguments contain sub- series summing over integers classified by the count of their prime factors. The core example is the Riemann zeta function ζ which accumulates the prime zeta function P1 plus what we shall define the almost-prime zeta functions Pk (Section 2). The central observation of this manuscript is that the almost-prime zeta functions are combinatorial sums over the prime zeta function (Section 3). Since earlier work by Cohen, Sebah and Gourdon has pointed at efficient numerical algorithms to compute P1, series over reciprocal almost-primes—which may suffer from slow convergence in their defining format—may be computed efficiently by reference to the P1. In consequence, any converging series over the positive integers which has a Taylor expansion in reciprocal powers of these integers splits into k-almost prime components. Section 4 illustrates this for the most basic formats. Number theory as such will not be advanced. The meromorphic landscape of the prime zeta functions as a function of their main variable appears to be more complicated than what is known for their host, the Riemann zeta function; so only some remarks on the calculation of first derivatives are dropped.
    [Show full text]
  • Université De Montréal Prime Number Races Tony Haddad
    Université de Montréal Prime Number Races par Tony Haddad Département de mathématiques et de statistique Faculté des arts et des sciences Mémoire présenté en vue de l’obtention du grade de Maître ès sciences (M.Sc.) en mathématiques Orientation mathématiques pures Août 2020 c Tony Haddad, 2020 Université de Montréal Faculté des arts et des sciences Ce mémoire intitulé Prime Number Races présenté par Tony Haddad a été évalué par un jury composé des personnes suivantes : Andrew Granville (président-rapporteur) Dimitris Koukoulopoulos (directeur de recherche) Matilde Lalín (membre du jury) Résumé Sous l’hypothèse de Riemann généralisée et l’hypothèse d’indépendance linéaire, Rubinstein et Sarnak ont prouvé que les valeurs de x > 1 pour lesquelles nous avons plus de nombres premiers de la forme 4n + 3 que de nombres premiers de la forme 4n + 1 en dessous de x ont une densité logarithmique d’environ 99,59 %. En général, l’étude de la différence #{p 6 x : p ∈ A} − #{p 6 x : p ∈ B} pour deux sous-ensembles de nombres premiers A et B s’appelle la course entre les nombres premiers de A et de B. Dans ce mémoire, nous cherchons ultimement à analyser d’un point de vue numérique et statistique la course entre les nombres premiers p tels que 2p + 1 est aussi premier (aussi appelés nombres premiers de Sophie Germain) et les nombres premiers p tels que 2p − 1 est aussi premier. Pour ce faire, nous présentons au préalable l’analyse de Rubinstein et Sarnak pour pouvoir repérer d’où vient le biais dans la course entre les nombres premiers 1 (mod 4) et les nombres premiers 3 (mod 4) et émettons une conjecture sur la distribution des nombres premiers de Sophie Germain.
    [Show full text]
  • Turbulent Candidates” That Note the Term 4290 Shown in Light Face Above Is in A(N), but Must Be Tested Via the Regular Counting Function to See If the Not in A244052
    Working Second Draft TurbulentMichael Thomas Candidates De Vlieger Abstract paper involve numbers whose factors have multiplicities that never approach 9, we will simply concatenate the digits, thus rendering This paper describes several conditions that pertain to the 75 = {0, 1, 2} as “012”. appearance of terms in oeis a244052, “Highly regular numbers a(n) defined as positions of records in a010846.” The sequence In multiplicity notation, a zero represents a prime totative q < is arranged in “tiers” T wherein each member n has equal values gpf(n) = a006530(n), the greatest prime factor of n. Indeed the multiplicity notation a054841(n) of any number n does not ex- ω(n) = oeis a001221(n). These tiers have primorials pT# = oeis a002110(T) as their smallest member. Each tier is divided into press the infinite series of prime totatives q > gpf(n). We could also write a054841(75) = 01200000…, with an infinite number of “levels” associated with integer multiples kpT# with 1 ≤ k < p(T + zeroes following the 2. Because it is understood that the notation 1). Thus all the terms of oeis a060735 are also in a244052. Mem- bers of a244052 that are not in a060735 are referred to as “turbu- “012” implies all the multiplicities “after” the 2 are 0. lent terms.” This paper focuses primarily on the nature of these Regulars. Consider two positive nonzero integers m and n. The terms. We give parameters for their likely appearance in tier T as number m is said to be “regular to” or “a regular of” n if and only if well as methods of efficiently constructing them.
    [Show full text]
  • Counting Restricted Partitions of Integers Into Fractions: Symmetry and Modes of the Generating Function and a Connection to Ω(T)
    Counting Restricted Partitions of Integers Into Fractions: Symmetry and Modes of the Generating Function and a Connection to !(t) Z. Hoelscher and E. Palsson∗ Abstract - Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions for integers smaller than the denominator form symmetric patterns. If the number of terms is restricted to h, then the nonzero terms of the generating function are unimodal, with the integer h having the most partitions. Such properties can be applied to a particular class of nonlinear Diophantine equations. We also examine partitions with even numerators. We prove that there are 2!(t) − 2 partitions of an integer t into fractions with the first x consecutive even integers for numerators and equal denominators of y, where 0 < y < x < t. We then use this to produce corollaries such as a Dirichlet series identity and an extension of the prime omega function to the complex plane, though this extension is not analytic everywhere. Keywords : integer partitions; restricted partitions; partitions into fractions Mathematics Subject Classification (2020) : 05A17 1 Introduction Integer partitions are a classic part of number theory. In the most general, unrestricted case, one seeks to express positive integers as the sum of smaller positive integers. Often the function p(n) is used to denote the count of the partitions of n. No simple formula for this is known, though a generating function can be written [1]. We note that a generating function is a formal power series where the exponent of x in each term refers to the number being partitioned, and the coefficient of x gives the number of partitions.
    [Show full text]
  • Aba Mbirika and Rita Post
    \Mathematics is the Queen of the Sciences, and Number Theory is the Queen of Mathematics." { Carl Friedrich Gauss (1777-1855) aBa Mbirika and Rita Post Spring 2019 LICENSE Creative Commons License (CC BY-NC-SA): This text, including the art and illustrations, are available under the Creative Commons license (CC BY-NC- SA), allowing anyone to reuse, revise, remix and redistribute the text. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-sa/4.0/ Section CONTENTS Page 3 Contents 0 Preliminaries9 0.1 Math Symbols Used In This Book . .9 0.2 What Is Number Theory and What Is Its Purpose? . 12 0.2.1 Some Beautiful Conjectures Awaiting Proofs . 22 0.2.2 Some Beautiful Results That Have Been Proven . 23 0.2.3 Number Theory Flourishes In Hollywood Films . 27 0.3 Examples of the Beauty and Power of Number Theory . 29 1 Proofs { An Introduction 37 1.1 Statements and Truth Tables . 38 1.2 Implications . 42 1.3 Predicates and Quantifiers . 46 1.4 Writing Formal Proofs . 56 1.5 Quick Review of Set Theory and Some Set Theory Proofs . 63 1.6 Butternut Squash . 69 2 Mathematical Induction 71 2.1 Motivation . 71 2.2 The Principle and Method of Mathematical Induction . 74 2.3 Exercises . 77 3 Pascal's Triangle and Binomial Coefficients 81 3.1 Motivation . 81 3.2 Constructing Pascal's Triangle . 84 3.3 16 Rows of Pascal's Triangle . 86 3.4 Binomial Coefficients . 87 3.5 Exercises . 88 4 Primes, Prime Factorization, and Some Consequences 93 4.1 Motivation .
    [Show full text]
  • A Handbook of Mathematical Discourse
    A Handbook of Mathematical Discourse Version 0.9 Charles Wells April 25, 2002 Charles Wells Professor Emeritus of Mathematics, Case Western Reserve University Affiliate Scholar, Oberlin College Address: 105 South Cedar Street Oberlin, OH 44074, USA Email: [email protected] Website: http://www.cwru.edu/artsci/math/wells/home.html Copyright c 2002 by Charles Wells Contents List of Words and Phrases Entries Preface Bibliography About this Handbook Citations Overview Point of view Index Coverage Left To Do Descriptive and Prescriptive Citations Citations needed Presentation Information needed Superscripted cross references TeX Problems Acknowledgments Things To Do contents wordlist index List of Words and Phrases abstract algebra arbitrary brace compartmentaliza- abstraction argument by bracket tion abuse of notation analogy but componentwise accumulation of argument calculate compositional attributes argument call compute action arity cardinality concept image ad-hoc article cases conceptual blend polymorphism assertion case conceptual affirming the assume category concept consequent assumption character conditional aha at most check assertion algebra attitudes circumflex conjunction algorithm addiction back formation citation connective algorithm bad at math classical category consciousness- alias bare delimiter closed under raising all barred arrow code example all notation codomain consequent always bar cognitive consider analogy behaviors dissonance constructivism and be collapsing contain angle bracket binary operation collective plural
    [Show full text]
  • Glossary of Mathematical Symbols
    Glossary of mathematical symbols A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics. The most basic symbols are the decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the Latin alphabet. The decimal digits are used for representing numbers through the Hindu–Arabic numeral system. Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. Letters are used for representing many other sort of mathematical objects. As the number of these sorts has dramatically increased in modern mathematics, the Greek alphabet and some Hebrew letters are also used. In mathematical formulas, the standard typeface is italic type for Latin letters and lower-case Greek letters, and upright type for upper case Greek letters. For having more symbols, other typefaces are also used, mainly boldface , script typeface (the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur , and blackboard bold (the other letters are rarely used in this face, or their use is unconventional). The use of Latin and Greek letters as symbols for denoting mathematical objects is not described in this article. For such uses, see Variable (mathematics) and List of mathematical constants. However, some symbols that are described here have the same shape as the letter from which they are derived, such as and .
    [Show full text]
  • Design a New Cryptosystem
    DESIGN A NEW CRYPTOSYSTEM Md. Shamim Hossain Biswas Publication Partner: IJSRP INC. 12/29/2019 http://dx.doi.org/10.29322/IJSRP.29.12.2019. Publication Partner: International Journal of Scientific and Research Publications (ISSN: 2250-3153) DESIGN A NEW CRYPTOSYSTEM M.S.H. Biswas Cryptosystem Md. Shamim Hossain Biswas Publishing Partner: IJSRP Inc. www.ijsrp.org http://dx.doi.org/10.29322/IJSRP.29.12.2019 Publication Partner: International Journal of Scientific and Research Publications (ISSN: 2250-3153) Preface Alhamdulillah, all praises to ALLAH (subhanahu wa ta′ala) who gives me the ability to complete this research work. I could not have finished my work if Almighty ALLAH did not make it possible. I would like to give special thanks to Dr. Touhid Bhuiyan (Heads and Professor, Department of Software Engineering, Daffodil International University) who provided opportunity to do this research. I am very grateful to Mr. Md. Maruf Hassan for his inspirational advices in cryptography lectures. I had motivated to cryptography by his inspiration and contented by reading cryptography. I am very grateful to Abu Shamim Aminur Razzaque for his rigorous encouragement and good academic advocating. I am very grateful to Mr. Md. Khaled Sohel who instructed me to find out research gaps during the decision period of my research activities. I would like to give thanks to M. Mostafa kamal who is trainer and writer of a number of books on English language for helping me during correction time of this research. I would like to vote of thanks to Dr. Md. Asraf Ali who gave some important instruction during this research.
    [Show full text]
  • The Math Encyclopedia of Smarandache Type Notions [Vol. I
    Vol. I. NUMBER THEORY MARIUS COMAN THE MATH ENCYCLOPEDIA OF SMARANDACHE TYPE NOTIONS Vol. I. NUMBER THEORY Educational Publishing 2013 Copyright 2013 by Marius Coman Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721 Peer-Reviewers: Dr. A. A. Salama, Faculty of Science, Port Said University, Egypt. Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Mohamed Eisa, Dept. of Computer Science, Port Said Univ., Egypt. EAN: 9781599732527 ISBN: 978-1-59973-252-7 2 INTRODUCTION About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozens of books and articles regarding math, physics, literature, philosophy). Being a globally recognized personality in both mathematics (there are countless functions and concepts that bear his name), it is natural that the volume of writings about his research is huge. What we try to do with this encyclopedia is to gather together as much as we can both from Smarandache’s mathematical work and the works of many mathematicians around the world inspired by the Smarandache notions. Because this is too vast to be covered in one book, we divide encyclopedia in more volumes. In this first volume of encyclopedia we try to synthesize his work in the field of number theory, one of the great Smarandache’s passions, a surfer on the ocean of numbers, to paraphrase the title of the book Surfing on the ocean of numbers – a few Smarandache notions and similar topics, by Henry Ibstedt.
    [Show full text]