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Sums Involving the Number of Distinct Prime Factors Function Rose-Hulman Undergraduate Mathematics Journal Volume 19 Issue 1 Article 8 Sums Involving the Number of Distinct Prime Factors Function Tanay Wakhare University of Maryland, College Park, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Part of the Number Theory Commons Recommended Citation Wakhare, Tanay (2018) "Sums Involving the Number of Distinct Prime Factors Function," Rose-Hulman Undergraduate Mathematics Journal: Vol. 19 : Iss. 1 , Article 8. Available at: https://scholar.rose-hulman.edu/rhumj/vol19/iss1/8 Sums Involving the Number of Distinct Prime Factors Function Cover Page Footnote I wrote this when I was 17, and looking back now I'm still amazed. This was my first ve er paper, and helped launch my research career -- and I couldn't have done it were it not for literally dozens of people around me. Many thanks go to my research supervisors at the National Institute of Standards and Technology: Guru Khalsa, Mark Stiles, Kyoung-Whan Kim, and Vivek Amin. I am also indebted to Howard Cohl, who took a frankly horrifying manuscript and turned it into a publishable paper. Without him this project wouldn't have gotten off the ground. Bruce Berndt, Krishna Alladi, and the anonymous referee also provided valuable feedback. Finally, many thanks go out to my old high school teachers -- William Wuu, Joseph Boettcher, Colleen Adams, Joshua Schuman, Jamie Andrews, and others -- who miraculously put up with me through the years. This article is available in Rose-Hulman Undergraduate Mathematics Journal: https://scholar.rose-hulman.edu/rhumj/ vol19/iss1/8 Rose- Hulman Undergraduate Mathematics Journal Sums Involving the Number of Distinct Prime Factors Function Tanay Wakhare a b Volume 19, No. 1, Fall, 2018 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 aQuince Orchard High School, Gaithersburg MD 20878 b [email protected] Center for Nanoscale Science and Technology, National Institute of scholar.rose-hulman.edu/rhumj Standards and Technology, Gaithersburg MD 20899 Rose-Hulman Undergraduate Mathematics Journal Volume 19, No. 1, Fall, 2018 Sums Involving the Number of Distinct Prime Factors Function Abstract. We nd closed form expressions for nite and innite sums that are weighted by !(n), where !(n) is the number of distinct prime factors of n. We then derive general convergence criteria for these series. The approach of this paper is to use the theory of symmetric functions to derive identities for the elementary symmetric functions, then apply these identities to arbitrary primes and values of multiplicative functions evaluated at primes. This allows us to reinterpret sums over symmetric polynomials as divisor sums and sums over the natural numbers. Page 2 RHIT Undergrad. Math. J., Vol. 19, No. 1 1 Introduction The number of distinct prime factors of a natural number, !(n), is a fundamental arithmetic quantity, and has been the subject of extensive study from Hardy and Ramanujan [5] to Erd®s and Kac [2]. We seek to expand the list of known series identities involving !(n), with potential applications to the asymptotics of !(n) and its higher moments !k(n). The prototype for our results is the Dirichlet series X !(n) = ζ(s)P (s); ns n2N where is the Riemann zeta function and is the prime zeta function, P 1 . ζ(s) P (s) P (s) := p ps Results such as this allow us to study the asymptotic behavior of the partial sums P . n≤x !(n) However, sometimes it is simpler to study smoothed asymptotics P , for some n≤x !(n)f(n) suitable arithmetic function f. Therefore, in this work we compute closed forms for Dirichlet series involving !(n)f(n), for any arbitary multiplicative weights f, and explicitly calculate many special cases. In the last section, we also lay out an extension for higher powers !(n)kf(n), and explicitly calculate some examples for k = 2. We also describe nite versions of the above identities, which involve divisor sums over !(n)f(n) for arbitary multiplicative f. This again helps expand our knowledge of the ! function, and the many of the special cases available here do not seem to be present anywhere else in the literature, perhaps because the easiest arithmetic functions to study are multiplicative, while !(n) is additive. Identities mixing both additive and multiplicative functions are not readily available using standard methods, so we use nonstandard symmetric function expansions to study sums involving both. The methods readily generalize, and can be applied to other additive functions. The paper is organized as follows. After standardizing some notation, we derive identities for the elemtary symmetric functions in Section 3. In Section 4, we explore divisor sums of !(n) weighted by other functions as corollaries of our results on symmetric polynomials. In Section 5, Dirichlet series of !(n) weighted by other functions are explored. We also compile an extensive list of corollaries, which may prove handy when considering exotic divisor sums. Finally, in Section 6 we present an extension to higher orders, and present a short summary in Section 7. 2 Background and notation We recall some denitions and background material which will be essential in later parts of this paper. Letting Q αi denote the prime factorization of , we have . n = 1≤i≤k pi n !(n) = k For example, !(5) = 1, !(10) = 2, and !(100) = !(2252) = 2. Recall that a multiplicative function satises f(nm) = f(n)f(m) for (n; m) = 1, where (n; m) gives the greatest common divisor of n and m. A completely multiplicative function satises f(nm) = f(n)f(m) for any n and m. We assume that f(1) = 1. An additive function satises f(nm) = f(n)+f(m) RHIT Undergrad. Math. J., Vol. 19, No. 1 Page 3 for (n; m) = 1, while a completely additive function satises f(nm) = f(n) + f(m) for any n and m. Now, we address some notation. We let P and Q denote sums and products over p ap p ap all primes , beginning with . We let P and Q denote sums and products p p = 2 pjn ap pjn ap over the distinct primes that divide a positive integer . Finally, we let P and Q n djn ad djn ad denote sums and products over the positive divisors of a positive integer n, including 1 and . For example, if , Q and Q . For the duration n n = 12 dj12 ad = a1a2a3a4a6a12 pj12 ap = a2a3 of this paper, the symbol N := f1; 2; 3 :::g will denote the set of positive integers and will be referred to as the natural numbers. We obey the convention that for s 2 C, s = σ + it with . We nally dene . σ; t 2 R Cα := fz 2 C : z 6= αg We also utilize the theory of Euler products. An Euler product, described in [8, (27.4.1- 2)], is the product form of a Dirichlet series. For any multiplicative function f we have [1, (11.8)] 1 X f(n) Y X f(pm) = (1 + a (p)) ; where a (p) := : (1) ns f;s f;s pms n2N p m=1 The left-hand side of the previous equation is a Dirichlet series, and the right-hand is an Euler product. Furthermore, letting s = σ + it, the abscissa of absolute convergence is the unique real number such that P f(n) absolutely converges if and only if . n2 ns σ > σa Some multiplicative functions whichN will be used in this paper include denotes the Euler totient function φ(n), which counts the number of natural numbers less than or equal to n which are coprime to n [8, (27.2.7)]. We will frequently use the multiplicative representation [8, (27.3.3)] Y 1 φ(n) = n 1 − : (2) p pjn A standard generalization of φ is Jordan's totient function [8, (27.3.4)], given by Y 1 J (n) := nk 1 − : (3) k pk pjn Using !(n), we can also dene the Möbius function, which gives the parity of the number of prime factors in a squarefree number, and is formally given by [8, (27.2.12)] 8 1; if n=1, <> µ(n) := 0; if n is non-squarefree, (4) :>(−1)!(n); if n is squarefree. 3 Factorization identities The main results of this paper are based on the following standard symmetric function identity. Page 4 RHIT Undergrad. Math. J., Vol. 19, No. 1 Proposition 3.1. Let and . Then n 2 N x1; : : : ; xn 2 C1 n ! n ! n Y X xi X k (1 − xi) = (−1) kek; (5) xi − 1 i=1 i=1 k=1 where ek = ek(x1; : : : ; xn) denotes the elementary symmetric polynomial dened as X (6) ek := xi1 xi2 ··· xik : 1≤i1<i2<···<ik≤n Proof. Following Macdonald [6, (Chapter 1)], we have the generating product n n Y X k k E(t) = (1 − txi) = (−1) t ek; (7) i=1 k=0 with . Taking a logarithmic derivative of for indeterminate yields xi 2 C E(t) t n X xi E0(t) = E(t) : (8) txi − 1 k=1 Taking t = 1 completes the proof. Corollary 3.2. Let and . Then n 2 N x1; : : : ; xn 2 C−1 n ! n ! n Y X xi X (1 + xi) = kek: (9) xi + 1 i=1 i=1 k=1 Proof. We repeat the previous argument in the proof of Proposition 3.1 but map t to −t before taking the logarithmic derivative. Note that by comparing coecients of tn we recover the Newton-Girard identities [6, (Chapter 1)], but these are in fact identities for the generating functions for various symmetric functions.
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