DOCSLIB.ORG

Scratch Work

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Scratch Work Introduction In the introduction, partially on the basis of several pages of scratch work, we tendered the conjecture that Ramanujan had devoted his last hours to cranks before his suffering became too intense to work on mathematics in the last four days of his brief life. In this short appendix we briefly examine some of the pages of scratch work. Most of the scratch work that we can identify pertains to calculations involving theta functions, cranks, or the partition function. We emphasize that these pages contain no exciting discoveries. The scratch work gives us glimpses of some of Ramanujan’s thoughts, but perhaps more importantly, the scratch work demonstrates the importance of calculations for Ramanujan. We discuss pages in the order in which they appear in [283]. Page 61 As discussed in Chapter 4, the first five tables are preliminary versions of the tables given on page 179. These tables are followed by seven lists of num- bers, three belonging to residue classes 1 modulo 3, three belonging to residue classes −1 modulo 3, and the last belonging to multiples of 3. However, gen- erally, neither the numbers nor the values of p(n) for these n belong to the requisite residue classes. The seven classes contain a total of 71 numbers, with the largest being 130. Below the tables is the quotient of (apparently) infinite products (q; q)2 ∞ . (q3; q3)∞ The scratch work at the bottom of the page contains several theta functions depicted by the first several terms of their infinite series representations, in particular, G.E. Andrews, B.C. Berndt, Ramanujan’s Lost Notebook: Part III, 403 DOI 10.1007/978-1-4614-3810-6, c Springer Science+Business Media New York 2012 404 Scratch Work 1 − q − q2 + q5 + q7 − q12 − q15 −···= f(−q), 1 − q2 − q7 + q13 + q23 − q33 − q48 + ···= f(−q2, −q7), 1 − q4 − q5 + q17 + q19 − q39 − q42 + ···= f(−q4, −q5). We also see 1 − q7 1+q2 + q4 + q6 − q7 + q8 − q9 + q10 + ···= . 1 − q2 Page 65=73, 66 Inexplicably, the publisher photocopied page 65 twice. Calculations on these pages appear to be related to cranks. Page 72 Some of the calculations appear to be related to cranks. Pages 74–77 These pages may be related to the generating functions in Chapters 2 and 3. In the upper left-hand corner of page 76, we find the expressions 1 − q2 − q3 + ··· (1 − q5)(1 − q10) , , (1 − q)2(1 − q4)2 (1 − q)(1 − q4) (1 − q5)(1 − q10) 1 − q − q4 + ··· , . (1 − q2)(1 − q5) (1 − q2)2(1 − q3)2 Page 78 Printed upside down, we find (1 − q7 − q8 + q29 + q31 −···)(1 − q5 − q10 + q25 + q35 −···) = f(−q7, −q8)f(−q5, −q10). Immediately following, we find 1 − q − q4 + q7 + q13 − q18 − q27 + q34 + q46 −···= f(−q, −q4). We remark that we have inserted −··· in each instance above. Page 82 405 Page 79 If the series 1 − q + q2 + q6 + q8 − q9 + q10 − q11 +2q12 were extended to infinity, it would equal 10 9 10 10 10 5 9 (q; q )∞(q ; q )∞(q ; q )∞ f(−q )f(−q, −q ) = . (q2; q5)∞(q3; q5)∞ f(−q2, −q3) In the middle of the page, we see 1+q + q2 + q3 +2q4 + q6 + q7 +2q8 + q9, which, if the summands were extended to infinity, would be equal to (q5; q5)2 f 3(−q5) ∞ = . (q; q5)∞(q4; q5)∞ f(−q, −q4) Page 80 This page appears to be related to calculations related to cranks. Reading sideways, we again see many functions identical to or similar to others that we have seen before, including 2 3 5 5 5 5 f(−q , −q ) (q ; q )∞ (q ; q )∞ 5 2 4 5 2 , 5 4 5 , 2 5 3 5 . (q; q )∞(q ; q )∞ (q; q )∞(q ; q )∞ (q ; q )∞(q ; q )∞ Page 81 This page may be related to Chapters 2 and 3. Page 82 At the top of the page we see 1+q − q2 − q5 + q7 + q12 − q15 − q22 + q26 + q35 − q40 − ... 0 ∞ = (−1)nqn(3n−1)/2 − (−1)nqn(3n−1)/2, n=−∞ n=1 which is a false theta function. 406 Scratch Work Pages 83–85 These pages are likely related to calculations involving cranks. Pages 86–89 These pages contain a table of the residue classes of p(n)forn from 1 to nearly 200. The first column is n, the second is p(n) (mod 2), the third is p(n) (mod 3), the fourth is p(n) (mod 5), the fifth is p(n) (mod 7), and the sixth is p(n) (mod 11). For the first 17 values of n, the residue of p(n) modulo 13, 17, 19, and 23 are also given. On the right side of page 87, Ramanujan also tabulates the number of values in each residue class modulo 2, 3, 5, and 7 for the first and second 50 values of n. On page 86, Ramanujan appears to have calculated residues modulo various n for some arithmetic function that we cannot identify. Page 213 This page contains a small amount of scratch work on Eisenstein series, but no theorem is offered. Location Guide For each page of Ramanujan’s lost notebook on which we have discussed or proved entries in this book, we provide below a list of those chapters or entries in which these pages are discussed. Page 18 Entries 2.1.1, 3.5.1 Page 19 Entries 2.1.4, 2.1.5, 3.6.1 Page 20 Entries 2.1.1, 2.1.2, 3.5.1 Page 54 Entries 9.1.1, 9.4.1, 9.4.2, 9.4.3, 9.4.4, 9.4.5 Page 58 Section 4.4 Page 59 Entries 3.1.1, 4.2.1, 4.2.3, Section 4.4 Page 61 Section 4.5, Scratch Work G.E. Andrews, B.C. Berndt, Ramanujan’s Lost Notebook: Part III, 407 DOI 10.1007/978-1-4614-3810-6, c Springer Science+Business Media New York 2012 408 Location Guide Page 63 Section 4.7 Page 64 Section 4.7 Page 65 Scratch Work Page 66 Scratch Work Page 70 Entry 3.7.1 Page 72–89 Scratch Work Pages 135–177 Chapter 5 Page 178 Chapter 7 Page 179 Entries 3.3.1, 3.4.1, Section 4.5 Page 180 Section 4.5 Page 181 Section 4.6 Page 182 Entries 6.5.1, 6.5.2, 6.5.3 Page 184 Entry 2.1.3 Location Guide 409 Page 189 Entry 6.2.1,Entry6.2.2,Entry6.3.1,Entry6.3.2, Entry 6.4.1,Entry6.4.2,Entry6.4.3 Page 207 Section 6.6 Page 208 Section 6.6 Page 213 Scratch Work Pages 236–237 Chapter 8 Pages 238–243 Chapter 5 Page 248 Section 6.6 Page 252 Section 6.6 Pages 280–312 Chapter 10 Page 326 Entry 6.6.1, Section 6.6 Page 331 Entries 6.6.2–6.6.5 Page 333 Section 6.6 Provenance Chapter 2 A.O.L. Atkin and H.P.F. Swinnerton-Dyer, [28] F.G. Garvan, [144], [146] Chapter 3 B.C. Berndt, H.H. Chan, S.H. Chan, and W.-C. Liaw, [62] Chapter 4 B.C. Berndt, H.H. Chan, S.H. Chan, and W.-C. Liaw, [63], [64] W.-C. Liaw, [216] Chapter 5 B.C. Berndt and K. Ono, [67] Chapter 6 B.C. Berndt, A.J. Yee, and J. Yi, [70] B.C. Berndt, C. Gugg, and S. Kim, [66] Chapter 7 J.M. Rushforth, [305], [306] Chapter 8 B.C. Berndt, G. Choi, Y.-S. Choi, H. Hahn, B. P. Yeap, A. J. Yee, H. Yesilyurt, and J. Yi, [65] B.C. Berndt and H. Yesilyurt, [73] H. Yesilyurt, [347], [348] G.E. Andrews, B.C. Berndt, Ramanujan’s Lost Notebook: Part III, 411 DOI 10.1007/978-1-4614-3810-6, c Springer Science+Business Media New York 2012 412 Provenance Chapter 9 A. Berkovich, F.G. Garvan, and H. Yesilyurt, [53] H.H. Chan, Z.-G. Liu, and S.T. Ng, [101] S.H. Son, [320] P. Xu, [345] Chapter 10 J.-L. Nicolas and G. Robin, [250] References 1. S. Ahlgren, Distribution of parity of the partition function in arithmetic pro- gressions, Indag. Math. 10 (1999), 173–181. 2. S. Ahlgren, Non-vanishing of the partition function modulo odd primes ,Math- ematika 46 (1999), 185–192. 3. S. Ahlgren, The partition function modulo composite integers M, Math. Ann. 318 (2000), 795–803. 4. S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan’s theta function, Proc. Amer. Math. Soc. 128 (2000), 1333–1338. 5. S. Ahlgren and M. Boylan, Arithmetic properties of the partition function, Invent. Math. 153 (2003), 487–502. 6. S. Ahlgren and M. Boylan, Coefficients of half-integral weight modular forms modulo , Math. Ann. 331 (2005), 219–241. 7. S. Ahlgren and K. Ono, Addition and counting: The arithmetic of partitions, Notices Amer. Math. Soc. 48 (2001), 978–984. 8. S. Ahlgren and S. Treneer, Rank generating functions as weakly holomorphic modular forms,ActaArith.133 (2008), 267–279. 9. L. Alaoglu and P. Erd˝os, On highly composite and similar numbers,Trans. Amer. Math. Soc. 56 (1944), 448–469. 10. A. Alaca, S. Alaca, M.F. Lemire, and K.S. Williams, Nineteen quatenary quadratic forms,ActaArith.130 (2007), 277–310. 11. A. Alaca, S. Alaca, and K.S. Williams, The simplest proof of Jacobi’s six squares theorem, Far East J. Math.
Recommended publications
  • On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function

    On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function

    mathematics Article On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function ŠtˇepánHubálovský 1,* and Eva Trojovská 2 1 Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic 2 Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +420-49-333-2704 Received: 3 October 2020; Accepted: 14 October 2020; Published: 16 October 2020 Abstract: Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk ≡ 0 (mod n). In this paper, we shall find all positive solutions of the Diophantine equation z(j(n)) = n, where j is the Euler totient function. Keywords: Fibonacci numbers; order of appearance; Euler totient function; fixed points; Diophantine equations MSC: 11B39; 11DXX 1. Introduction Let (Fn)n≥0 be the sequence of Fibonacci numbers which is defined by 2nd order recurrence Fn+2 = Fn+1 + Fn, with initial conditions Fi = i, for i 2 f0, 1g. These numbers (together with the sequence of prime numbers) form a very important sequence in mathematics (mainly because its unexpectedly and often appearance in many branches of mathematics as well as in another disciplines). We refer the reader to [1–3] and their very extensive bibliography. We recall that an arithmetic function is any function f : Z>0 ! C (i.e., a complex-valued function which is defined for all positive integer).
  • Hypergeometric Functions and Feynman Diagrams 3

    Hypergeometric Functions and Feynman Diagrams 3

    Hypergeometric Functions and Feynman Diagrams Mikhail Kalmykov, Vladimir Bytev, Bernd Kniehl, Sven-Olaf Moch, Bennie Ward, and Scott Yost Abstract The relationship between Feynman diagrams and hypergeometric functions is discussed. Special attention is devoted to existing techniquesfor the constructionof the Y-expansion. As an example, we present a detailed discussion of the construction of the Y-expansion of the Appell function 3 around rational values of parameters via an iterative solution of differential equations. As a by-product,we have found that the one-loop massless pentagon diagram in dimension 3 = 3 2n is not expressible in terms of multiple polylogarithms. Another interesting example− is the Puiseux-type solution involving a differential operator generated by a hypergeometric function of three variables. The holonomic properties of the # hypergeometric functions are briefly discussed. Mikhail Kalmykov JINR, Dubna, Russia, e-mail: [email protected] Vladimir Bytev BLTP JINR, 141980 Dubna, Moscow region, Russia e-mail: [email protected] Bernd A. Kniehl, II. Institut fuer Theoretische Physik, Universitaet Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany e-mail: [email protected] Sven-Olaf Moch II. Institut fuer Theoretische Physik, Universitaet Hamburg, Luruper Chaussee 149, 22761 arXiv:2012.14492v3 [hep-th] 22 Jan 2021 Hamburg, Germany e-mail: [email protected] B.F.L. Ward Department of Physics, Baylor University, One Bear Place, # 97316, Waco, TX 76798-7316, USA e- mail: BFL{\protect\relax$\@@underline{\hbox{\}}\mathsurround\z@$\relax}[email protected] S.A. Yost Department of Physics, The Citadel, 171 Moultrie St., Charleston, SC 29409, USA e-mail: [email protected] 1 2 Authors Suppressed Due to Excessive Length 1 Introduction Recent interest in the analytical properties of Feynman diagrams has been motivated by processes at the LHC.
  • Arxiv:Math/0201082V1

    Arxiv:Math/0201082V1

    2000]11A25, 13J05 THE RING OF ARITHMETICAL FUNCTIONS WITH UNITARY CONVOLUTION: DIVISORIAL AND TOPOLOGICAL PROPERTIES. JAN SNELLMAN Abstract. We study (A, +, ⊕), the ring of arithmetical functions with unitary convolution, giving an isomorphism between (A, +, ⊕) and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell- Everett[4] between the ring (A, +, ·) of arithmetical functions with Dirichlet convolution and the power series ring C[[x1,x2,x3,... ]] on countably many variables. We topologize it with respect to a natural norm, and shove that all ideals are quasi-finite. Some elementary results on factorization into atoms are obtained. We prove the existence of an abundance of non-associate regular non-units. 1. Introduction The ring of arithmetical functions with Dirichlet convolution, which we’ll denote by (A, +, ·), is the set of all functions N+ → C, where N+ denotes the positive integers. It is given the structure of a commutative C-algebra by component-wise addition and multiplication by scalars, and by the Dirichlet convolution f · g(k)= f(r)g(k/r). (1) Xr|k Then, the multiplicative unit is the function e1 with e1(1) = 1 and e1(k) = 0 for k> 1, and the additive unit is the zero function 0. Cashwell-Everett [4] showed that (A, +, ·) is a UFD using the isomorphism (A, +, ·) ≃ C[[x1, x2, x3,... ]], (2) where each xi corresponds to the function which is 1 on the i’th prime number, and 0 otherwise. Schwab and Silberberg [9] topologised (A, +, ·) by means of the norm 1 arXiv:math/0201082v1 [math.AC] 10 Jan 2002 |f| = (3) min { k f(k) 6=0 } They noted that this norm is an ultra-metric, and that ((A, +, ·), |·|) is a valued ring, i.e.
  • Fixed Points of Certain Arithmetic Functions

    Fixed Points of Certain Arithmetic Functions

    FIXED POINTS OF CERTAIN ARITHMETIC FUNCTIONS WALTER E. BECK and RUDOLPH M. WAJAR University of Wisconsin, Whitewater, Wisconsin 53190 IWTRODUCTIOW Perfect, amicable and sociable numbers are fixed points of the arithemetic function L and its iterates, L (n) = a(n) - n, where o is the sum of divisor's function. Recently there have been investigations into functions differing from L by 1; i.e., functions L+, Z.,, defined by L + (n)= L(n)± 1. Jerrard and Temperley [1] studied the existence of fixed points of L+ and /._. Lai and Forbes [2] conducted a computer search for fixed points of (LJ . For earlier references to /._, see the bibliography in [2]. We consider the analogous situation using o*, the sum of unitary divisors function. Let Z.J, Lt, be arithmetic functions defined by L*+(n) = o*(n)-n±1. In § 1, we prove, using parity arguments, that L* has no fixed points. Fixed points of iterates of L* arise in sets where the number of elements in the set is equal to the power of/.* in question. In each such set there is at least one natural number n such that L*(n) > n. In § 2, we consider conditions n mustsatisfy to enjoy the inequality and how the inequality acts under multiplication. In particular if n is even, it is divisible by at least three primes; if odd, by five. If /? enjoys the inequality, any multiply by a relatively prime factor does so. There is a bound on the highest power of n that satisfies the inequality. Further if n does not enjoy the inequality, there are bounds on the prime powers multiplying n which will yield the inequality.
  • Some New Results for Humbert's Double

    Some New Results for Humbert's Double

    Palestine Journal of Mathematics Vol. 8(2)(2019) , 146–158 © Palestine Polytechnic University-PPU 2019 SOME NEW RESULTS FOR HUMBERT’S DOUBLE HYPERGEOMETRIC SERIES 2 AND φ2 Ghazala Yasmin and Abdulghani Muhyi Communicated by P.K. Banerji MSC 2010 Classifications: Primary 33B15, 33C20; Secondary 33C70. Keywords and phrases: Gamma Function, Generalized Hypergeometric Function, Kummer’s Theorem, Generalized (Wright) Hypergeometric Function, Humbert’s Functions, Lavoie-Trottier Integral Formula. Abstract. This article aims to establish new expressions of Humbert’s functions 2 and φ2 with the help of the generalization of Kummer’s summation theorem. Some special cases are also obtained. Further, we present some interesting integrals involving the Humbert’s functions, which are expressed in terms of generalized (Wright) hypergeometric function. Also, some special cases were considered as an application of the presented integral formulas. 1 Introduction The theory of hypergeometric functions has a vast popularity and great interest. The usefulness of this functions lies in the fact that the solutions of many applied problems involving partial dif- ferential equations are obtainable with the help of such hypergeometric functions. For instance, the energy absorbed by some non ferromagnetic conductor sphere contained in an internal mag- netic field can be calculated with the help of such functions [12, 15]. Multi-variable hyper- geometric functions are used in physical and quantum chemical applications as well [14, 19]. Especially, many problems in gas dynamics lead to those of degenerate second order partial differential equations, which are then solvable in terms of multiple hypergeometric functions. Further, in the investigation of the boundary value problems for these partial differential equa- tions, we need decompositions for multivariate hypergeometric functions in terms of simpler hypergeometric functions of the Gauss and Humbert types.
  • On Some New Arithmetic Functions Involving Prime Divisors and Perfect Powers

    On Some New Arithmetic Functions Involving Prime Divisors and Perfect Powers

    On some new arithmetic functions involving prime divisors and perfect powers. Item Type Article Authors Bagdasar, Ovidiu; Tatt, Ralph-Joseph Citation Bagdasar, O., and Tatt, R. (2018) ‘On some new arithmetic functions involving prime divisors and perfect powers’, Electronic Notes in Discrete Mathematics, 70, pp.9-15. doi: 10.1016/ j.endm.2018.11.002 DOI 10.1016/j.endm.2018.11.002 Publisher Elsevier Journal Electronic Notes in Discrete Mathematics Download date 30/09/2021 07:19:18 Item License http://creativecommons.org/licenses/by/4.0/ Link to Item http://hdl.handle.net/10545/623232 On some new arithmetic functions involving prime divisors and perfect powers Ovidiu Bagdasar and Ralph Tatt 1,2 Department of Electronics, Computing and Mathematics University of Derby Kedleston Road, Derby, DE22 1GB, United Kingdom Abstract Integer division and perfect powers play a central role in numerous mathematical results, especially in number theory. Classical examples involve perfect squares like in Pythagora’s theorem, or higher perfect powers as the conjectures of Fermat (solved in 1994 by A. Wiles [8]) or Catalan (solved in 2002 by P. Mih˘ailescu [4]). The purpose of this paper is two-fold. First, we present some new integer sequences a(n), counting the positive integers smaller than n, having a maximal prime factor. We introduce an arithmetic function counting the number of perfect powers ij obtained for 1 ≤ i, j ≤ n. Along with some properties of this function, we present the sequence A303748, which was recently added to the Online Encyclopedia of Integer Sequences (OEIS) [5]. Finally, we discuss some other novel integer sequences.
  • Problem Solving and Recreational Mathematics

    Problem Solving and Recreational Mathematics

    Problem Solving and Recreational Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Summer 2012 Chapters 1–44 August 1 Monday 6/25 7/2 7/9 7/16 7/23 7/30 Wednesday 6/27 *** 7/11 7/18 7/25 8/1 Friday 6/29 7/6 7/13 7/20 7/27 8/3 ii Contents 1 Digit problems 101 1.1 When can you cancel illegitimately and yet get the cor- rectanswer? .......................101 1.2 Repdigits.........................103 1.3 Sortednumberswithsortedsquares . 105 1.4 Sumsofsquaresofdigits . 108 2 Transferrable numbers 111 2.1 Right-transferrablenumbers . 111 2.2 Left-transferrableintegers . 113 3 Arithmetic problems 117 3.1 AnumbergameofLewisCarroll . 117 3.2 Reconstruction of multiplicationsand divisions . 120 3.2.1 Amultiplicationproblem. 120 3.2.2 Adivisionproblem . 121 4 Fibonacci numbers 201 4.1 TheFibonaccisequence . 201 4.2 SomerelationsofFibonaccinumbers . 204 4.3 Fibonaccinumbersandbinomialcoefficients . 205 5 Counting with Fibonacci numbers 207 5.1 Squaresanddominos . 207 5.2 Fatsubsetsof [n] .....................208 5.3 Anarrangementofpennies . 209 6 Fibonacci numbers 3 211 6.1 FactorizationofFibonaccinumbers . 211 iv CONTENTS 6.2 TheLucasnumbers . 214 6.3 Countingcircularpermutations . 215 7 Subtraction games 301 7.1 TheBachetgame ....................301 7.2 TheSprague-Grundysequence . 302 7.3 Subtraction of powers of 2 ................303 7.4 Subtractionofsquarenumbers . 304 7.5 Moredifficultgames. 305 8 The games of Euclid and Wythoff 307 8.1 ThegameofEuclid . 307 8.2 Wythoff’sgame .....................309 8.3 Beatty’sTheorem . 311 9 Extrapolation problems 313 9.1 Whatis f(n + 1) if f(k)=2k for k =0, 1, 2 ...,n? . 313 1 9.2 Whatis f(n + 1) if f(k)= k+1 for k =0, 1, 2 ...,n? .
  • On Product Partitions of Integers

    On Product Partitions of Integers

    Canad. Math. Bull.Vol. 34 (4), 1991 pp. 474-479 ON PRODUCT PARTITIONS OF INTEGERS V. C. HARRIS AND M. V. SUBBARAO ABSTRACT. Let p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers > 2, the order of the factors in the product being irrelevant, with p*(\ ) = 1. For any integer d > 1 let dt = dxll if d is an /th power, and = 1, otherwise, and let d = 11°^ dj. Using a suitable generating function for p*(ri) we prove that n^,, dp*(n/d) = np*inK 1. Introduction. The well-known partition function p(n) stands for the number of unrestricted partitions of n, that is, the number of ways of expressing a given positive integer n as the sum of one or more positive integers, the order of the parts in the partition being irrelevant. In contrast to this, we consider here the function p*(n), which denotes the number of ways of expressing n as the product of positive integers > 2, the order of the factors in the product being irrelevant. For example, /?*(12) = 4, since 12 can be expressed in positive integers > 2 as a product in these and only these ways: 12,6 • 2, 4-3, 3-2-2. We may say that/?(n) denotes the number of sum partitions and p*(n) the number of product partitions of n. We note that the number of product partitions of n = p\axpiai.. .pkak in standard form is independent of the particular primes involved; for example, /?*(12) = p*(22 • 3) = p*(p\p2) for every choice of distinct primes p\ and/?2- For computing it is usually convenient to let pj be they'th prime; for easily ordering the divisors in increasing order it suffices to take/?2 > P\a\P?> > Pia]P2ai, etc.
  • Primes in Arithmetical Progression

    Primes in Arithmetical Progression

    Colby College Digital Commons @ Colby Honors Theses Student Research 2019 Primes in Arithmetical Progression Edward C. Wessel Colby College Follow this and additional works at: https://digitalcommons.colby.edu/honorstheses Part of the Analysis Commons, and the Number Theory Commons Colby College theses are protected by copyright. They may be viewed or downloaded from this site for the purposes of research and scholarship. Reproduction or distribution for commercial purposes is prohibited without written permission of the author. Recommended Citation Wessel, Edward C., "Primes in Arithmetical Progression" (2019). Honors Theses. Paper 935. https://digitalcommons.colby.edu/honorstheses/935 This Honors Thesis (Open Access) is brought to you for free and open access by the Student Research at Digital Commons @ Colby. It has been accepted for inclusion in Honors Theses by an authorized administrator of Digital Commons @ Colby. Primes in Arithmetical Progression Edward (Teddy) Wessel April 2019 Contents 1 Message to the Reader 3 2 Introduction 3 2.1 Primes . 3 2.2 Euclid . 3 2.3 Euler and Dirichlet . 4 2.4 Shapiro . 4 2.5 Formal Statement . 5 3 Arithmetical Functions 6 3.1 Euler’s Totient Function . 6 3.2 The Mobius Function . 6 3.3 A Relationship Between j and m ................ 7 3.4 The Mangoldt Function . 8 4 Dirichlet Convolution 9 4.1 Definition . 10 4.2 Some Basic Properties of Dirichlet Multiplication . 10 4.3 Identity and Inverses within Dirichlet Multiplication . 11 4.4 Multiplicative Functions and their Relationship with Dirich- let Convolution . 13 4.5 Generalized Convolutions . 15 4.6 Partial Sums of Dirichlet Convolution .
  • On the Parity of the Number of Multiplicative Partitions and Related Problems

    On the Parity of the Number of Multiplicative Partitions and Related Problems

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 11, November 2012, Pages 3793–3803 S 0002-9939(2012)11254-7 Article electronically published on March 15, 2012 ON THE PARITY OF THE NUMBER OF MULTIPLICATIVE PARTITIONS AND RELATED PROBLEMS PAUL POLLACK (Communicated by Ken Ono) Abstract. Let f(N) be the number of unordered factorizations of N,where a factorization is a way of writing N as a product of integers all larger than 1. For example, the factorizations of 30 are 2 · 3 · 5, 5 · 6, 3 · 10, 2 · 15, 30, so that f(30) = 5. The function f(N), as a multiplicative analogue of the (additive) partition function p(N), was first proposed by MacMahon, and its study was pursued by Oppenheim, Szekeres and Tur´an, and others. Recently, Zaharescu and Zaki showed that f(N) is even a positive propor- tion of the time and odd a positive proportion of the time. Here we show that for any arithmetic progression a mod m,thesetofN for which f(N) ≡ a(mod m) possesses an asymptotic density. Moreover, the density is positive as long as there is at least one such N. For the case investigated by Zaharescu and Zaki, we show that f is odd more than 50 percent of the time (in fact, about 57 percent). 1. Introduction Let f(N) be the number of unordered factorizations of N,whereafactorization of N is a way of writing N as a product of integers larger than 1. For example, f(12) = 4, corresponding to 2 · 6, 2 · 2 · 3, 3 · 4, 12.
  • Randell Heyman School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia Randell@Unsw.Edu.Au

    Randell Heyman School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia [email protected]

    #A67 INTEGERS 19 (2019) CARDINALITY OF A FLOOR FUNCTION SET Randell Heyman School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia [email protected] Received: 5/13/19, Revised: 9/16/19, Accepted: 12/22/19, Published: 12/24/19 Abstract Fix a positive integer X. We quantify the cardinality of the set X/n : 1 n X . We discuss restricting the set to those elements that are prime,{bsemiprimec or similar. } 1. Introduction Throughout we will restrict the variables m and n to positive integer values. For any real number X we denote by X its integer part, that is, the greatest integer b c that does not exceed X. The most straightforward sum of the floor function is related to the divisor summatory function since X = 1 = ⌧(n), n nX6X ⌫ nX6X k6XX/n nX6X where ⌧(n) is the number of divisors of n. From [2, Theorem 2] we infer X = X log X + X(2γ 1) + O X517/1648+o(1) , n − nX6X ⌫ ⇣ ⌘ where γ is the Euler–Mascheroni constant, in particular γ 0.57722. ⇡ Recent results have generalized this sum to X f , n nX6X ✓ ⌫◆ where f is an arithmetic function (see [1], [3] and [4]). In this paper we take a di↵erent approach by examining the cardinality of the set X S(X) := m : m = for some n X . n ⇢ ⌫ Our main results are as follows. INTEGERS: 19 (2019) 2 Theorem 1. Let X be a positive integer. We have S(X) = p4X + 1 1. | | − j k Theorem 2.
  • A New Proof of a Reduction Formula for the Appell Series F3 Due to Bailey∗

    A New Proof of a Reduction Formula for the Appell Series F3 Due to Bailey∗

    FACTA UNIVERSITATIS (NIS)ˇ Ser. Math. Inform. Vol. 34, No 5 (2019), 849–854 https://doi.org/10.22190/FUMI1905849M A NEW PROOF OF A REDUCTION FORMULA FOR THE APPELL SERIES F3 DUE TO BAILEY∗ Gradimir V. Milovanovi´cand Arjun K. Rathie c 2019 by University of Niˇs, Serbia | Creative Commons Licence: CC BY-NC-ND Abstract. In this short note, we provide a new proof of an interesting and useful reduction formula for the Appell series F3 due to Bailey [On the sum of a terminating 3F2(1), Quart. J. Math. Oxford Ser. (2) 4 (1953), 237–240]. Keywords: Appell series, Humbert series, Whipple summation theorem, reduction formula, special functions 1. Introduction Hypergeometric series and many their generalizations, including multiple series, play an important role in the applied mathematics and mathematical physics. In this short note we are interested only for multiple hypergeometric series of Appell type, precisely for Appell series F3, which is defined by (cf. [2, 3, 6]) ∞ ∞ ′ ′ m n ′ ′ (a)m(a )n(b)m(b )n w z (1.1) F3(a,a ,b,b ; c; w,z)= m n m=0 n=0 (c) + · m! n! X X for w < 1, z < 1, c =0, 1, 2,... | | | | 6 − − As usual (λ)k is the well known Pochhammer symbol (or the shifted factorial or the raised factorial, since (1)n = n!) defined by Γ(λ + k) (λ)k = = λ(λ + 1) (λ + k 1). Γ(λ) ··· − A survey on multiple hypergeometric series of Appell type with a rich list of references has been recently published by Schlosser [10].