DOCSLIB.ORG

The Handbook of Mathematical Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Handbook of Mathematical Functions Chapter 27 Functions of Number Theory T. M. Apostol1 Notation 638 Additive Number Theory 644 27.1 Special Notation ............. 638 27.13 Functions ................. 644 27.14 Unrestricted Partitions .......... 645 Multiplicative Number Theory 638 Applications 647 27.2 Functions ................. 638 27.15 Chinese Remainder Theorem ....... 647 27.3 Multiplicative Properties ......... 640 27.16 Cryptography ............... 647 27.4 Euler Products and Dirichlet Series ... 640 27.17 Other Applications ............ 647 27.5 Inversion Formulas ............ 641 Computation 648 27.6 Divisor Sums ............... 641 27.18 Methods of Computation: Primes .... 648 27.7 Lambert Series as Generating Functions . 641 27.19 Methods of Computation: Factorization . 648 27.8 Dirichlet Characters ............ 642 27.20 Methods of Computation: Other Number- 27.9 Quadratic Characters ........... 642 Theoretic Functions ............ 649 27.10 Periodic Number-Theoretic Functions .. 642 27.21 Tables ................... 649 27.22 Software .................. 649 27.11 Asymptotic Formulas: Partial Sums ... 643 27.12 Asymptotic Formulas: Primes ...... 644 References 649 1California Institute of Technology, Pasadena, California. Acknowledgments: The author thanks Basil Gordon for comments on an earlier draft, and David Bressoud for providing xx27.12, 27.18, 27.19, and 27.22. Copyright c 2009 National Institute of Standards and Technology. All rights reserved. 637 638 Functions of Number Theory π(x) that counts the number of primes not exceeding x. Notation It can be expressed as a sum over all primes p ≤ x: X 27.2.2 π(x) = 1: 27.1 Special Notation p≤x Gauss and Legendre conjectured that π(x) is asymp- (For other notation see pp. xiv and 873.) totic to x= log x as x ! 1: d; k; m; n positive integers (unless otherwise x 27.2.3 π(x) ∼ : indicated). log x d j n d divides n. (See Gauss(1863, Band II, pp. 437{477) and Legendre (m; n) greatest common divisor of m; n. If (1808, p. 394).) (m; n) = 1, m and n are called relatively This result, first proved in Hadamard(1896) and prime, or coprime. de la Vall´eePoussin(1896a,b), is known as the prime (d1; : : : ; dn) greatest common divisor of d1; : : : ; dn. number theorem. An equivalent form states that the P Q djn, djn sum, product taken over divisors of n. nth prime pn (when the primes are listed in increasing P (m;n)=1 sum taken over m, 1 ≤ m ≤ n and m order) is asymptotic to n log n as n ! 1: relatively prime to n. 27.2.4 pn ∼ n log n: p; p1; p2;::: prime numbers (or primes): integers (See also x27.12.) Other examples of number-theoretic (> 1) with only two positive integer functions treated in this chapter are as follows. divisors, 1 and the number itself. ( P , Q sum, product extended over all primes. 1 1; n = 1; p p 27.2.5 = x; y real numbers. n 0; n > 1: P Pbxc n≤x n=1. X k φk(n) = m ; log x natural logarithm of x, written as ln x in 27.2.6 (m;n)=1 other chapters. the sum of the kth powers of the positive integers m ≤ n ζ(s) Riemann zeta function; see x25.2(i). that are relatively prime to n. (njP ) Jacobi symbol; see x27.9. (njp) Legendre symbol; see x27.9. 27.2.7 φ(n) = φ0(n): This is the number of positive integers ≤ n that are relatively prime to n; φ(n) is Euler's totient. If (a; n) = 1, then the Euler{Fermat theorem states Multiplicative Number Theory that 27.2.8 aφ(n) ≡ 1 (mod n); and if φ(n) is the smallest positive integer f such that 27.2 Functions af ≡ 1 (mod n), then a is a primitive root mod n. The 2 φ(n) 27.2(i) Definitions φ(n) numbers a; a ; : : : ; a are relatively prime to n and distinct (mod n). Such a set is a reduced residue Functions in this section derive their properties from the system modulo n. fundamental theorem of arithmetic, which states that X d(n) = 1 every integer n > 1 can be represented uniquely as a 27.2.9 product of prime powers, djn is the number of divisors of n and is the divisor func- ν(n) tion. It is the special case k = 2 of the function dk(n) Y ar 27.2.1 n = pr ; that counts the number of ways of expressing n as the r=1 product of k factors, with the order of factors taken into where p ; p ; : : : ; p are the distinct prime factors of account. 1 2 ν(n) X σ (n) = dα; n, each exponent ar is positive, and ν(n) is the number 27.2.10 α of distinct primes dividing n.(ν(1) is defined to be 0.) djn Euclid's Elements (Euclid(1908, Book IX, Proposition is the sum of the αth powers of the divisors of n, where 20)) gives an elegant proof that there are infinitely many the exponent α can be real or complex. Note that primes. Tables of primes (x27.21) reveal great irregular- σ0(n) = d(n). ity in their distribution. They tend to thin out among X J (n) = 1; the large integers, but this thinning out is not com- 27.2.11 k pletely regular. There is great interest in the function ((d1;:::;dk);n)=1 27.2 Functions 639 is the number of k-tuples of integers ≤ n whose greatest This is Liouville's function. common divisor is relatively prime to n. This is Jordan's 27.2.14 Λ(n) = log p, n = pa, function. Note that J1(n) = φ(n). a In the following examples, a1; : : : ; aν(n) are the ex- where p is a prime power with a ≥ 1; otherwise ponents in the factorization of n in (27.2.1). Λ(n) = 0. This is Mangoldt's function. 8 1; n = 1; <> ν(n) 27.2.12 µ(n) = (−1) ; a1 = a2 = ··· = aν(n) = 1; 27.2(ii) Tables :>0; otherwise: This is the M¨obiusfunction. Table 27.2.1 lists the first 100 prime numbers pn. Ta- ( ble 27.2.2 tabulates the Euler totient function φ(n), the 1; n = 1; divisor function d(n) (= σ (n)), and the sum of the 27.2.13 0 λ(n) = a +···+a (−1) 1 ν(n) ; n > 1: divisors σ(n) (= σ1(n)), for n = 1(1)52. Table 27.2.1: Primes. n pn pn+10 pn+20 pn+30 pn+40 pn+50 pn+60 pn+70 pn+80 pn+90 1 2 31 73 127 179 233 283 353 419 467 2 3 37 79 131 181 239 293 359 421 479 3 5 41 83 137 191 241 307 367 431 487 4 7 43 89 139 193 251 311 373 433 491 5 11 47 97 149 197 257 313 379 439 499 6 13 53 101 151 199 263 317 383 443 503 7 17 59 103 157 211 269 331 389 449 509 8 19 61 107 163 223 271 337 397 457 521 9 23 67 109 167 227 277 347 401 461 523 10 29 71 113 173 229 281 349 409 463 541 Table 27.2.2: Functions related to division. n φ(n) d(n) σ(n) n φ(n) d(n) σ(n) n φ(n) d(n) σ(n) n φ(n) d(n) σ(n) 1 1 1 1 14 6 4 24 27 18 4 40 40 16 8 90 2 1 2 3 15 8 4 24 28 12 6 56 41 40 2 42 3 2 2 4 16 8 5 31 29 28 2 30 42 12 8 96 4 2 3 7 17 16 2 18 30 8 8 72 43 42 2 44 5 4 2 6 18 6 6 39 31 30 2 32 44 20 6 84 6 2 4 12 19 18 2 20 32 16 6 63 45 24 6 78 7 6 2 8 20 8 6 42 33 20 4 48 46 22 4 72 8 4 4 15 21 12 4 32 34 16 4 54 47 46 2 48 9 6 3 13 22 10 4 36 35 24 4 48 48 16 10 124 10 4 4 18 23 22 2 24 36 12 9 91 49 42 3 57 11 10 2 12 24 8 8 60 37 36 2 38 50 20 6 93 12 4 6 28 25 20 3 31 38 18 4 60 51 32 4 72 13 12 2 14 26 12 4 42 39 24 4 56 52 24 6 98 640 Functions of Number Theory 27.3 Multiplicative Properties Euler product of the series. If f(n) is completely multi- plicative, then each factor in the product is a geometric Except for ν(n), Λ(n), pn, and π(x), the functions in series and the Euler product becomes x27.2 are multiplicative, which means f(1) = 1 and 1 X Y −1 27.3.1 f(mn) = f(m)f(n), (m; n) = 1. 27.4.2 f(n) = (1 − f(p)) : If f is multiplicative, then the values f(n) for n > 1 n=1 p are determined by the values at the prime powers. Euler products are used to find series that generate Specifically, if n is factored as in (27.2.1), then many functions of multiplicative number theory. The completely multiplicative function f(n) = n−s gives the ν(n) Euler product representation of the Riemann zeta func- Y ar 27.3.2 f(n) = f(pr ): tion ζ(s)(x25.2(i)): r=1 1 In particular, X Y 27.4.3 ζ(s) = n−s = (1 − p−s)−1, <s > 1.
Load more

Recommended publications
  • Euler Product Asymptotics on Dirichlet L-Functions 3
    EULER PRODUCT ASYMPTOTICS ON DIRICHLET L-FUNCTIONS IKUYA KANEKO Abstract. We derive the asymptotic behaviour of partial Euler products for Dirichlet L-functions L(s,χ) in the critical strip upon assuming only the Generalised Riemann Hypothesis (GRH). Capturing the behaviour of the partial Euler products on the critical line is called the Deep Riemann Hypothesis (DRH) on the ground of its importance. Our result manifests the positional relation of the GRH and DRH. If χ is a non-principal character, we observe the √2 phenomenon, which asserts that the extra factor √2 emerges in the Euler product asymptotics on the central point s = 1/2. We also establish the connection between the behaviour of the partial Euler products and the size of the remainder term in the prime number theorem for arithmetic progressions. Contents 1. Introduction 1 2. Prelude to the asymptotics 5 3. Partial Euler products for Dirichlet L-functions 6 4. Logarithmical approach to Euler products 7 5. Observing the size of E(x; q,a) 8 6. Appendix 9 Acknowledgement 11 References 12 1. Introduction This paper was motivated by the beautiful and profound work of Ramanujan on asymptotics for the partial Euler product of the classical Riemann zeta function ζ(s). We handle the family of Dirichlet L-functions ∞ L(s,χ)= χ(n)n−s = (1 χ(p)p−s)−1 with s> 1, − ℜ 1 p X Y arXiv:1902.04203v1 [math.NT] 12 Feb 2019 and prove the asymptotic behaviour of partial Euler products (1.1) (1 χ(p)p−s)−1 − 6 pYx in the critical strip 0 < s < 1, assuming the Generalised Riemann Hypothesis (GRH) for this family.
    [Show full text]
  • Limiting Values and Functional and Difference Equations †
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 February 2020 doi:10.20944/preprints202002.0245.v1 Peer-reviewed version available at Mathematics 2020, 8, 407; doi:10.3390/math8030407 Article Limiting values and functional and difference equations † N. -L. Wang 1, P. Agarwal 2,* and S. Kanemitsu 3 1 College of Applied Mathematics and Computer Science, ShangLuo University, Shangluo, 726000 Shaanxi, P.R. China; E-mail:[email protected] 2 Anand International College of Engineering, Near Kanota, Agra Road, Jaipur-303012, Rajasthan, India 3 Faculty of Engrg Kyushu Inst. Tech. 1-1Sensuicho Tobata Kitakyushu 804-8555, Japan; [email protected] * Correspondence: [email protected] † Dedicated to Professor Dr. Yumiko Hironaka with great respect and friendship Version February 17, 2020 submitted to Journal Not Specified Abstract: Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function. Keywords: limit values; modular relation; Lerch zeta-function; Hurwitz zeta-function; Laurent coefficients MSC: 11F03; 01A55; 40A30; 42A16 1. Introduction There have appeared enormous amount of papers on the Laurent coefficients of a large class of zeta-, L- and special functions.
    [Show full text]
  • Of the Riemann Hypothesis
    A Simple Counterexample to Havil's \Reformulation" of the Riemann Hypothesis Jonathan Sondow 209 West 97th Street New York, NY 10025 [email protected] The Riemann Hypothesis (RH) is \the greatest mystery in mathematics" [3]. It is a conjecture about the Riemann zeta function. The zeta function allows us to pass from knowledge of the integers to knowledge of the primes. In his book Gamma: Exploring Euler's Constant [4, p. 207], Julian Havil claims that the following is \a tantalizingly simple reformulation of the Rie- mann Hypothesis." Havil's Conjecture. If 1 1 X (−1)n X (−1)n cos(b ln n) = 0 and sin(b ln n) = 0 na na n=1 n=1 for some pair of real numbers a and b, then a = 1=2. In this note, we first state the RH and explain its connection with Havil's Conjecture. Then we show that the pair of real numbers a = 1 and b = 2π=ln 2 is a counterexample to Havil's Conjecture, but not to the RH. Finally, we prove that Havil's Conjecture becomes a true reformulation of the RH if his conclusion \then a = 1=2" is weakened to \then a = 1=2 or a = 1." The Riemann Hypothesis In 1859 Riemann published a short paper On the number of primes less than a given quantity [6], his only one on number theory. Writing s for a complex variable, he assumes initially that its real 1 part <(s) is greater than 1, and he begins with Euler's product-sum formula 1 Y 1 X 1 = (<(s) > 1): 1 ns p 1 − n=1 ps Here the product is over all primes p.
    [Show full text]
  • The Mobius Function and Mobius Inversion
    Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Number Theory Mathematics via Primary Historical Sources (TRIUMPHS) Winter 2020 The Mobius Function and Mobius Inversion Carl Lienert Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_number Part of the Curriculum and Instruction Commons, Educational Methods Commons, Higher Education Commons, Number Theory Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits ou.y Recommended Citation Lienert, Carl, "The Mobius Function and Mobius Inversion" (2020). Number Theory. 12. https://digitalcommons.ursinus.edu/triumphs_number/12 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Number Theory by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. The Möbius Function and Möbius Inversion Carl Lienert∗ January 16, 2021 August Ferdinand Möbius (1790–1868) is perhaps most well known for the one-sided Möbius strip and, in geometry and complex analysis, for the Möbius transformation. In number theory, Möbius’ name can be seen in the important technique of Möbius inversion, which utilizes the important Möbius function. In this PSP we’ll study the problem that led Möbius to consider and analyze the Möbius function. Then, we’ll see how other mathematicians, Dedekind, Laguerre, Mertens, and Bell, used the Möbius function to solve a different inversion problem.1 Finally, we’ll use Möbius inversion to solve a problem concerning Euler’s totient function.
    [Show full text]
  • Prime Factorization, Zeta Function, Euler Product
    Analytic Number Theory, undergrad, Courant Institute, Spring 2017 http://www.math.nyu.edu/faculty/goodman/teaching/NumberTheory2017/index.html Section 2, Euler products version 1.2 (latest revision February 8, 2017) 1 Introduction. This section serves two purposes. One is to cover the Euler product formula for the zeta function and prove the fact that X p−1 = 1 : (1) p The other is to develop skill and tricks that justify the calculations involved. The zeta function is the sum 1 X ζ(s) = n−s : (2) 1 We will show that the sum converges as long as s > 1. The Euler product formula is Y −1 ζ(s) = 1 − p−s : (3) p This formula expresses the fact that every positive integers has a unique rep- resentation as a product of primes. We will define the infinite product, prove that this one converges for s > 1, and prove that the infinite sum is equal to the infinite product if s > 1. The derivative of the zeta function is 1 X ζ0(s) = − log(n) n−s : (4) 1 This formula is derived by differentiating each term in (2), as you would do for a finite sum. We will prove that this calculation is valid for the zeta sum, for s > 1. We also can differentiate the product (3) using the Leibnitz rule as 1 though it were a finite product. In the following calculation, r is another prime: 8 9 X < d −1 X −1= ζ0(s) = 1 − p−s 1 − r−s ds p : r6=p ; 8 9 X <h −2i X −1= = − log(p)p−s 1 − p−s 1 − r−s p : r6=p ; ( ) X −1 = − log(p) p−s 1 − p−s ζ(s) p ( 1 !) X X ζ0(s) = − log(p) p−ks ζ(s) : (5) p 1 If we divide by ζ(s), the sum on the right is a sum over prime powers (numbers n that have a single p in their prime factorization).
    [Show full text]
  • 4 L-Functions
    Further Number Theory G13FNT cw ’11 4 L-functions In this chapter, we will use analytic tools to study prime numbers. We first start by giving a new proof that there are infinitely many primes and explain what one can say about exactly “how many primes there are”. Then we will use the same tools to proof Dirichlet’s theorem on primes in arithmetic progressions. 4.1 Riemann’s zeta-function and its behaviour at s = 1 Definition. The Riemann zeta function ζ(s) is defined by X 1 1 1 1 1 ζ(s) = = + + + + ··· . ns 1s 2s 3s 4s n>1 The series converges (absolutely) for all s > 1. Remark. The series also converges for complex s provided that Re(s) > 1. 2 4 P 1 Example. From the last chapter, ζ(2) = π /6, ζ(4) = π /90. But ζ(1) is not defined since n diverges. However we can still describe the behaviour of ζ(s) as s & 1 (which is my notation for s → 1+, i.e. s > 1 and s → 1). Proposition 4.1. lim (s − 1) · ζ(s) = 1. s&1 −s R n+1 1 −s Proof. Summing the inequality (n + 1) < n xs dx < n for n > 1 gives Z ∞ 1 1 ζ(s) − 1 < s dx = < ζ(s) 1 x s − 1 and hence 1 < (s − 1)ζ(s) < s; now let s & 1. Corollary 4.2. log ζ(s) lim = 1. s&1 1 log s−1 Proof. Write log ζ(s) log(s − 1)ζ(s) 1 = 1 + 1 log s−1 log s−1 and let s & 1.
    [Show full text]
  • Riemann Hypothesis and Random Walks: the Zeta Case
    Riemann Hypothesis and Random Walks: the Zeta case Andr´e LeClaira Cornell University, Physics Department, Ithaca, NY 14850 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 1 its L-function is valid to the right of the critical line <(s) > 2 , 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 1-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. arXiv:1601.00914v3 [math.NT] 23 May 2017 a [email protected] 1 I. INTRODUCTION There are many generalizations of Riemann's zeta function to other Dirichlet series, which are also believed to satisfy a Riemann Hypothesis. A common opinion, based largely on counterexamples, is that the L-functions for which the Riemann Hypothesis is true enjoy both an Euler product formula and a functional equation. However a direct connection between these properties and the Riemann Hypothesis has not been formulated in a precise manner.
    [Show full text]
  • Appendices A. Quadratic Reciprocity Via Gauss Sums
    1 Appendices We collect some results that might be covered in a first course in algebraic number theory. A. Quadratic Reciprocity Via Gauss Sums A1. Introduction In this appendix, p is an odd prime unless otherwise specified. A quadratic equation 2 modulo p looks like ax + bx + c =0inFp. Multiplying by 4a, we have 2 2ax + b ≡ b2 − 4ac mod p Thus in studying quadratic equations mod p, it suffices to consider equations of the form x2 ≡ a mod p. If p|a we have the uninteresting equation x2 ≡ 0, hence x ≡ 0, mod p. Thus assume that p does not divide a. A2. Definition The Legendre symbol a χ(a)= p is given by 1ifa(p−1)/2 ≡ 1modp χ(a)= −1ifa(p−1)/2 ≡−1modp. If b = a(p−1)/2 then b2 = ap−1 ≡ 1modp,sob ≡±1modp and χ is well-defined. Thus χ(a) ≡ a(p−1)/2 mod p. A3. Theorem a The Legendre symbol ( p ) is 1 if and only if a is a quadratic residue (from now on abbre- viated QR) mod p. Proof.Ifa ≡ x2 mod p then a(p−1)/2 ≡ xp−1 ≡ 1modp. (Note that if p divides x then p divides a, a contradiction.) Conversely, suppose a(p−1)/2 ≡ 1modp.Ifg is a primitive root mod p, then a ≡ gr mod p for some r. Therefore a(p−1)/2 ≡ gr(p−1)/2 ≡ 1modp, so p − 1 divides r(p − 1)/2, hence r/2 is an integer. But then (gr/2)2 = gr ≡ a mod p, and a isaQRmodp.
    [Show full text]
  • First Year Progression Report
    First Year Progression Report Steven Charlton 12 June 2013 Contents 1 Introduction 1 2 Polylogarithms 2 2.1 Definitions . .2 2.2 Special Values, Functional Equations and Singled Valued Versions . .3 2.3 The Dedekind Zeta Function and Polylogarithms . .5 2.4 Iterated Integrals and Their Properties . .7 2.5 The Hopf algebra of (Motivic) Iterated Integrals . .8 2.6 The Polygon Algebra . .9 2.7 Algebraic Structures on Polygons . 11 2.7.1 Other Differentials on Polygons . 11 2.7.2 Operadic Structure of Polygons . 11 2.7.3 A VV-Differential on Polygons . 13 3 Multiple Zeta Values 16 3.1 Definitions . 16 3.2 Relations and Transcendence . 17 3.3 Algebraic Structure of MZVs and the Standard Relations . 19 3.4 Motivic Multiple Zeta Values . 21 3.5 Dimension of MZVs and the Hoffman Elements . 23 3.6 Results using Motivic MZVs . 24 3.7 Motivic DZVs and Eisenstein Series . 27 4 Conclusion 28 References 29 1 Introduction The polylogarithms Lis(z) are an important and frequently occurring class of functions, with applications throughout mathematics and physics. In this report I will give an overview of some of the theory surrounding them, and the questions and lines of research still open to investigation. I will introduce the polylogarithms as a generalisation of the logarithm function, by means of their Taylor series expansion, multiple polylogarithms will following by considering products. This will lead to the idea of functional equations for polylogarithms, capturing the symmetries of the function. I will give examples of such functional equations for the dilogarithm.
    [Show full text]
  • Overpartitions and Functions from Multiplicative Number Theory
    U.P.B. Sci. Bull., Series A, Vol. 83, Iss. 3, 2021 ISSN 1223-7027 OVERPARTITIONS AND FUNCTIONS FROM MULTIPLICATIVE NUMBER THEORY Mircea Merca1 Let α and β be two nonnegative integers such that β < α. For an arbi- trary sequence fa g of complex numbers, we consider the generalized Lambert series n n>1 P αn−β αn−β n 1 anq =(1 − q ) in order to investigate linear combinations of the form P > S(αk − β; n)ak, where S(k; n) is the total number of non-overlined parts equal k>1 to k in all the overpartitions of n. The general nature of the numbers an allows us to provide connections between overpartitions and functions from multiplicative number theory. Keywords: partitions, Lambert series, theta series MSC2020: 11P81 05A19. 1. Introduction A partition of a positive integer n is a sequence of positive integers whose sum is n. The order of the summands is unimportant when writing the partitions of n, but for consistency, a partition of n will be written with the summands in a nonincreasing order [2]. An overpartition of a positive integer n is a partition of n in which the first occurrence of a part of each size may be overlined or not [6]. Let p(n) denote the number of overpartitions of an integer n. For example, p(3) = 8 because there are 8 overpartitions of 3: (3); (3)¯ ; (2; 1); (2¯; 1); (2; 1)¯ ; (2¯; 1)¯ ; (1; 1; 1); (1¯; 1; 1): Since the overlined parts form a partition into distinct parts and the non-overlined parts form an ordinary partition, the generating function for overpartitions is given by 1 X (−q; q)1 p(n)qn = : (q; q) n=0 1 Here and throughout this paper, we use the following customary q-series notation: ( 1; for n = 0, (a; q)n = (1 − a)(1 − aq) ··· (1 − aqn−1); for n > 0; (a; q)1 = lim (a; q)n: n!1 Because the infinite product (a; q)1 diverges when a 6= 0 and jqj > 1, whenever (a; q)1 appears in a formula, we shall assume jqj < 1.
    [Show full text]
  • Generating Special Arithmetic Functions by Lambert Series Factorizations
    Volume 14, Number 1, Pages 31{45 ISSN 1715-0868 GENERATING SPECIAL ARITHMETIC FUNCTIONS BY LAMBERT SERIES FACTORIZATIONS MIRCEA MERCA AND MAXIE D. SCHMIDT Abstract. In this collaborative article, we summarize the known use- ful and interesting results and formulas we have discovered so far. Sum- marizing results from two related articles by Merca and Schmidt, we arrive at so-termed Lambert series factorization theorems. We unify the matrix representations that underlie two of our separate papers, and which commonly arise in identities involving partition functions and other functions generated by Lambert series. We provide a number of properties and open problems related to the inverse matrix entries de- fined in Schmidt's article and the Euler partition function p(n) which we prove through our new results unifying the expansions of the Lambert series factorization theorems within this article. 1. Introduction 1.1. Lambert series factorization theorems. We consider recurrence relations and matrix equations related to Lambert series expansions of the form [4, x27.7] [1, x17.10] n X anq X (1.1) = b qm; jqj < 1; 1 − qn m n≥1 m≥1 + + for prescribed arithmetic functions a : Z ! C and b : Z ! C where bm = P djm ad. There are many well-known Lambert series for special arithmetic functions of the form in (1.1). Examples include the following series where µ(n) denotes the M¨obiusfunction, φ(n) denotes Euler's phi function, σα(n) denotes the generalized sum of divisors function, λ(n) denotes Liouville's function, Λ(n) denotes von Mangoldt's function, !(n) defines the number of distinct primes dividing n, and Jt(n) is Jordan's totient function for a fixed Received by the editors October 9, 2017, and in revised form January 3, 2018.
    [Show full text]
  • Rapidly Converging Formulae for $\Zeta (4K\Pm 1) $
    RAPIDLY CONVERGING FORMULAE FOR ζ(4k 1) ± SHUBHO BANERJEE AND BLAKE WILKERSON Abstract. We provide rapidly converging formulae for the Riemann zeta L s n − n function at odd integers using the Lambert series q(s)= n∞=1 n q /(1 q ), s = −(4k ± 1). Our main formula for ζ(4k − 1) convergesP at rate of about √15π 4π e− per term, and the formula for ζ(4k + 1), at the rate of e− per ≈ π3√15 term. For example, the first order approximation yields ζ(3) 100 + √15π 9 4 √15π 10 e− h 4 + √15 sinh( 2 )i which has an error only of order 10− . 1. Introduction In a recent paper [3] we analyzed the q 1− asymptotic behavior of a Lambert series of the type | | → ∞ nsqnx (1.1) Lq(s, x)= , s C, 1 qn ∈ n=1 X − with q < 1 and x > 0. At x = 1, the Lambert series Lq(s, x) is the generating | | function for the divisor function σs(n), the sum of the sth powers of divisors of an integer n [1]: ∞ n (1.2) σs(n) q = Lq(s, 1). n=1 X For odd positive integer s values, the Lambert series is related to the Eisentein series 4k ∞ n arXiv:1803.03291v1 [math.NT] 9 Mar 2018 (1.3) E2k(q)=1 σ2k 1(n) q , − B k − 2 n=1 X that are modular in nature [2]. In this paper we focus on the Lambert series at negative odd integer values of s = (4k 1), k = 1, 2, 3, ..
    [Show full text]