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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]
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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 SUMS INVOLVING THE NUMBER OF DISTINCT PRIME FACTORS FUNCTION
TANAY WAKHARE
Abstract. We nd closed form expressions for nite and in nite sums that are weighted by ω(n), where ω(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.
1. Introduction The number of distinct prime factors of a natural number, ω(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 ω(n), with potential applications to the asymptotics of ω(n) and its higher moments ωk(n). The prototype for our results is the Dirichlet series X ω(n) = ζ(s)P (s), ns n∈N where is the Riemann zeta function and is the prime zeta function, P 1 . Results such as this ζ(s) P (s) P (s) := p ps allow us to study the asymptotic behavior of the partial sums P . However, sometimes it is simpler to n≤x ω(n) study smoothed asymptotics P , for some suitable arithmetic function . Therefore, in this work we n≤x ω(n)f(n) f compute closed forms for Dirichlet series involving ω(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 ω(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 ω(n)f(n) for arbitary multiplicative f. This again helps expand our knowledge of the ω 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 ω(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 ω(n) weighted by other functions as corollaries of our results on symmetric polynomials. In Section 5, Dirichlet series of ω(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 de nitions and background material which will be essential in later parts of this paper. Letting Q αi denote the prime factorization of , we have . For example, , , and n = 1≤i≤k pi n ω(n) = k ω(5) = 1 ω(10) = 2 ω(100) = ω(2252) = 2. Recall that a multiplicative function satis es 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 satis es f(nm) = f(n)f(m) for any n and m. We assume that f(1) = 1. An additive function satis es f(nm) = f(n) + f(m) for (n, m) = 1, while a completely additive function satis es 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 all primes , beginning p ap p ap p with . We let P and Q denote sums and products over the distinct primes that divide a positive p = 2 p|n ap p|n ap integer . Finally, we let P and Q denote sums and products over the positive divisors of a positive n d|n ad d|n ad
MSC(2010): 11C08;11M06;11N99;11Z05. Keywords: Symmetric Polynomials; Analytic Number Theory; Distinct Prime Factors; Dirichlet Series; Divisor Sums; Multiplicative Number Theory. SUMS INVOLVING THE NUMBER OF DISTINCT PRIME FACTORS FUNCTION integer , including and . For example, if , Q and Q . For the n 1 n n = 12 d|12 ad = a1a2a3a4a6a12 p|12 ap = a2a3 duration of this paper, the symbol N := {1, 2, 3 ...} will denote the set of positive integers and will be referred to as the natural numbers. We obey the convention that for s ∈ C, s = σ + it with σ, t ∈ R. We nally de ne . Cα := {z ∈ C : z 6= α} 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)] ∞ X f(n) Y X f(pm) (2.1) = (1 + a (p)) , where a (p) := . ns f,s f,s pms n∈N 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 n∈N ns converges if and only if σ > σa. Some multiplicative functions which 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 (2.2) φ(n) = n 1 − . p p|n A standard generalization of φ is Jordan's totient function [8, (27.3.4)], given by Y 1 (2.3) J (n) := nk 1 − . k pk p|n Using ω(n), we can also de ne 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)] 1, if n=1, (2.4) µ(n) := 0, if n is non-squarefree, (−1)ω(n), if n is squarefree. 3. Factorization identities The main results of this paper are based on the following standard symmetric function identity. Proposition 1. Let and . Then n ∈ N x1, . . . , xn ∈ C1 n ! n ! n Y X xi X k (3.1) (1 − xi) = (−1) kek, xi − 1 i=1 i=1 k=1 where ek = ek(x1, . . . , xn) denotes the elementary symmetric polynomial de ned as (3.2) X ek := xi1 xi2 ··· xik .
1≤i1 Note that by comparing coe cients 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. 4. Divisor sums The following theorems are obtained by reinterpreting (3.1). We consider divisor sums of a multiplicative function f weighted by ω(n) and µ(n) or |µ(n)| = µ2(n). Throughout, f(p) will refer to the value of f evaluated at any prime . We also de ne P as if , where is any function, not necessarily multiplicative, because p p|n g(p) 0 n = 1 g 1 has no distinct prime factors. Under this convention, the theorems in this section also hold for n = 1. Theorem 3. Let f(n) be a multiplicative function de ned such that f(p) 6= 1 for any prime p that divides n. Then X Y X f(p) (4.1) µ(d)ω(d)f(d) = (1 − f(p)) . f(p) − 1 d|n p|n p|n Proof. First take any squarefree natural number , so that Qω(n) by the fundamental theorem of arithmetic. n n = i=1 pi We then let xi = f(pi) in (3.1), such that each xi is an arithmetic function evaluated at each distinct prime that divides n. We can evaluate ek, yielding X ek = f(pi1 )f(pi2 ) ··· f(pik ). 1≤i1 Since f is multiplicative and each pi is coprime to the others by de nition, we have X ek = f(pi1 pi2 ··· pik ). 1≤i1 Now we can regard ek as the sum of f(n) evaluated at the divisors of n with k prime factors, since each term in ek trivially has k prime factors, and every possible product of k primes that divide n is included in ek. This is equivalent to partitioning the divisors of n based on their number of distinct prime factors. Then (3.1) transforms into ω(n) ω(n) Y X f(pi) X ω(d) (1 − f(pi)) = ω(d)(−1) f(d). f(pi) − 1 i=1 i=1 d|n Each divisor d is squarefree since n is squarefree, so we can replace (−1)ω(d) by µ(d), where µ(d) is the Möbius function which is de ned by (2.4). Rewriting the product and sum over pi, 1 ≤ i ≤ ω(n), as a product and sum over p gives (4.1) for squarefree numbers. However, we can immediately see that if n is non-squarefree, µ(d) eliminates any non-squarefree divisors on the left-hand side. Meanwhile, the right-hand side is evaluated over the distinct primes that divide n so changing the multiplicities of these primes will not aect the sum in any way. Therefore (4.1) is valid for all n ∈ N, which completes the proof. Corollary 4. Let f(n) be a multiplicative function, n ∈ N, and f(p) 6= 1 for any prime p that divides n. Then X X X f(p) (4.2) µ(d)ω(d)f(d) = µ(d)f(d) . f(p) − 1 d|n d|n p|n Proof. A general theorem for any multiplicative function f(n), found in [1, (2.18)], is that X Y (4.3) µ(d)f(d) = (1 − f(p)) . d|n p|n Substituting this relation into (4.1) completes the proof. Theorem 5. Let f(n) be a multiplicative function, n ∈ N, and f(p) 6= −1 for any prime p that divides n. Then X Y X f(p) (4.4) |µ(d)|ω(d)f(d) = (1 + f(p)) . 1 + f(p) d|n p|n p|n Proof. We follow the proof of Theorem 3 but substitute xi = f(pi) into (3.5) instead of (3.1). Hence, we only have to take the divisor sum over squarefree divisors without multiplying by (−1)ω(d). We do this by multiplying the divisor sums by |µ(d)|, the characteristic function of the squarefree numbers. This completes the proof. SUMS INVOLVING THE NUMBER OF DISTINCT PRIME FACTORS FUNCTION Corollary 6. Let f(n) be a multiplicative function with f(1) = 1, f(p) 6= −1, and n ∈ N. Then X X X f(p) (4.5) |µ(d)|ω(d)f(d) = |µ(d)|f(d) . 1 + f(p) d|n d|n p|n Proof. We substitute f(n) = µ(n)g(n) into (4.3), where g(n) is multiplicative, ensuring that f(n) is also multiplica- tive. Noting that µ2(n) = |µ(n)| since the Möbius function only takes values of ±1 and 0 yields X Y |µ(d)|g(d) = (1 + g(p)) . d|n p|n Mapping g to f to maintain consistent notation and substituting into (4.4) completes the proof. Specializing f(n) yields an assortment of new formulae involving convolutions with ω(n). Below we use a variety of functions f(n) along with (4.1), (4.2), and (4.4) to nd new expressions for divisor sums involving ω(n). Corollary 7. Let n ∈ N. Then X (4.6) |µ(d)|ω(d) = ω(n)2ω(n)−1. d|n Proof. Substituting into (4.4) gives P Q P 1 Since the product and sum f(n) = 1 d|n |µ(d)|ω(d) = p|n 2 p|n 2 . on the right-hand side are over the distinct primes that divide n, each is evaluated ω(n) times. This simpli es to P ω(n) ω(n) . d|n |µ(d)|ω(d) = 2 2 Corollary 8. Let n ∈ N. Then X nk X 1 (4.7) µ(d)ω(d) = J (n) . d k 1 − pk d|n p|n Proof. Substituting 1 into (4.1) gives f(n) = nk X µ(d)ω(d) Y 1 X 1 (4.8) = 1 − . dk pk 1 − pk d|n p|n p|n Using (2.3) to see that Q 1 Jk(n) and substituting this relation into (4.8) completes the proof. p|n(1 − pk ) = nk Corollary 9. Let n ∈ N. Then k (4.9) X n J2k(n) X 1 |µ(d)|ω(d) = k . d Jk(n) 1 + p d|n p|n Proof. Substituting 1 into (4.4) gives f(n) = nk X |µ(d)|ω(d) Y 1 X 1 = 1 + . dk pk 1 + pk d|n p|n p|n Now we complete the proof by using (2.3) to see that 2k Q 1 Y 1 n p|n(1 − p2k ) 1 J2k(n) 1 + = = . k 2k Q 1 k p n (1 − ) n Jk(n) p|n p|n pk Corollary 10. Let n ∈ N with n squarefree. Then k (4.10) X k J2k(n) X p ω(d)d = k . Jk(n) 1 + p d|n p|n SUMS INVOLVING THE NUMBER OF DISTINCT PRIME FACTORS FUNCTION Proof. Substituting f(n) = nk into (4.4) gives X Y X pk (4.11) |µ(d)|ω(d)dk = (1 + pk) . 1 + pk d|n p|n p|n Now if we let be squarefree, then Q . We also know that (2.3) transforms into n n = p|n p Q k Y 1 p|n(p − 1) J (n) = nk 1 − = nk . k pk Q pk p|n p|n However we now have k Q k, so Q k . Then n = p|n p Jk(n) = p|n(p − 1) Q (p2k − 1) 2k (4.12) J2k(n) p|n Y (p − 1) Y k = Q k = k = (p + 1). Jk(n) (p − 1) (p − 1) p|n p|n p|n Letting n be a squarefree natural number in (4.4), we can also eliminate |µ(d)| from the sum on the left-hand side since every divisor of n will already be squarefree. Substituting (4.12) into (4.11) completes the proof. 5. Infinite sums The propositions (3.1) and (3.5) hold for a nite number of elements xi. However, we can take the limit i → ∞ to extend this sum. This enables us to nd closed form product expressions for Dirichlet series, which are described in [8, (27.4.4)], of the form P ω(n)f(n) for many commonly encountered multiplicative functions f(n). n∈N ns We introduce the prime zeta function [3, (0.1)], denoted by P (s). We de ne it by X 1 P (s) := , ps p and note that it converges for <(s) > 1. It is an analog of the Riemann zeta function, described in [8, (25.2.1)], with the sum taken over prime numbers instead of all natural numbers. We also de ne the shifted prime zeta function P (s, a) as X 1 P (s, a) := , ps + a p such that P (s, 0) = P (s). Lemma 11. Let a ∈ C and |a| < 2. Then P(s,a) converges absolutely if and only if s ∈ C, <(s) > 1. Proof. Consider a ∈ C. Now if <(s) = 1, we prove that P (s, a) diverges if |a| < 2. This implies that P (s, a) diverges for all s such that <(s) < 1 for these values of a. To prove divergence, it su ces to let a ∈ R since R ⊂ C. We let s = 1, a ∈ R, and 2 > a ≥ 0. Peeling o the i = 1 term, where p1 = 2, we have pi+1 − pi ≥ 2 > a ≥ 0 since the minimum prime gap for primes not equal to 2 and 3 is 2. This means ∞ ∞ ∞ X 1 1 X 1 1 X 1 1 5 P (1, a) = = + ≥ + = + P (1) − . p + a 2 + a p + a 2 + a p 2 + a 6 i=1 i i=2 i i=2 i+1 However, [3] gives us that P (1) diverges. Now we let s = 1 and −2 < a ≤ 0. Then by the direct comparison test, 1 1 . This means is bounded below by and the sum diverges. The restriction is because p+a > p P (s, a) P (1) a > −2 there is a simple pole if s = 1 and a = −2. Now we let <(s) > 1, a ∈ C, and 0 ≤ |a| < 2. By the triangle inequality, |ps| = |ps + a − a| ≤ |ps + a| + | − a| = |ps + a| + |a| so then |ps + a| ≥ |ps| − |a| = p<(s) − |a| > 0, since p ∈ N and p ≥ 2. Then we can bound P (s, a) so that X 1 X 1 |P (s, a)| ≤ ≤ . ps + a p<(s) − |a| p p We have now reduced the consideration of a complex s to considering a real s. Letting <(s) = σ for convenience, we have that σ σ . This allows us to form the bound pi − pi−1 ≥ 2 > |a| ≥ 0 ∞ ∞ ∞ X 1 1 X 1 1 X 1 1 P (s, a) ≤ = + ≤ + = + P (σ). pσ − |a| 2σ − |a| pσ − |a| 2σ − |a| pσ 2σ − |a| i=1 i i=2 i i=2 i−1 We know that P (s) converges for <(s) = σ > 1, so P (s, a) will also converge for 0 ≤ |a| < 2, completing the proof. SUMS INVOLVING THE NUMBER OF DISTINCT PRIME FACTORS FUNCTION Lemma 12. Let with . Then P pk converges absolutely if and only if . a, s, k ∈ C |a| < 2 p ps+a <(s) > max (1, 1 + <(k)) Proof. First we assume <(k) ≤ 0. Then k k X p X p X 1 ≤ ≤ , ps + a ps + a ps + a p p p which by (11) converges absolutely if and only if s > 1. Now we assume <(k) > 0. Then k X p X 1 ≤ . ps + a ps−k + ap−k p p Since a < 2 < 2, the conditions for (11) are satis ed and the series converges absolutely if and only if <(s−k) > 1. pk 2 Combining the two requirements for <(s) completes the proof. Our results on in nite series are based on a certain symmetric polynomial evaluation, which is a re nement of the Euler product/Dirichlet series duality ∞ X f(n) Y X f(pm) = (1 + a (p)) , where a (p) := . ns f,s f,s pms n∈N p m=1 In particular, given the evaluation of ek from Lemma 13, we can formally consider the generating product ∞ ∞ X k Y E(t) = ekt = (1 + xit), k≥0 i=1 set t = 1 and xi = af,s(pi), and rearrange the resulting series as ∞ ∞ Y X X f(n) X f(n) (1 + a (p )) = = . f,s i ns ns i=1 k=0 n≥1 n≥1 ω(n)=k This is precisely the de nition of an Euler product, though we have ignored questions of convergence in this comparison. The following lemma precisely encodes the correspondence above, and is the key element in the proof of Theorem 14. th Lemma 13. Let 1 ≤ i < ∞, i ∈ , pi denote the i prime, f denote any multiplicative function, and xi = m N P∞ f(pi ) ms . Furthermore, let k ∈ and m=1 pi N (5.1) Sk := {n ∈ N : ω(n) = k}, so that is the set of natural numbers with distinct prime factors. If P f(n) has an abscissa of absolute Sk k s n∈N n convergence σa, then X X f(n) (5.2) e := x x ··· x = , k i1 i2 ik ns 1≤i1 i1 i2 ik X f(pi )f(pi ) ··· f(pi ) e = 1 2 k . k s p i1 p i2 ··· p ik 1≤i1 Here i varies because it goes over every single power of p which is present in xi. Since xi is the sum of a subsequence of P f(n) , it will also converge absolutely for and any rearrangement of its terms will not change the s σ > σa n∈N n value of the sum. Since f is multiplicative and each pi is coprime to the others by de nition, we have f p i1 p i2 ··· p ik X i1 i2 ik e = . k s p i1 p i2 ··· p ik 1≤i1 −1 Replacing by f(p) in (5.3) and simplifying completes the proof. We also note that P f(p) af,s(p) 1 − s − 1 s p n∈N p is the sum over a subsequence of P f(n) , so it will also converge absolutely for and we can simplify our s σ > σa n∈N n convergence criterion. Theorem 16. Let f(n) be a multiplicative function. Let P |µ(n)|f(n) and P f(p) both converge absolutely n∈N ns p ps+f(p) for . Assume that for all prime and such that we have s. Then σ > σa p s ∈ C σ > σa f(p) 6= −p ! ! X |µ(n)|ω(n)f(n) X |µ(n)|f(n) X f(p) (5.5) = , ns ns ps + f(p) n∈N n∈N p which converges absolutely for σ > σa. Proof. To begin, we note that a product of multiplicative functions is also multiplicative. Letting f(n) = |µ(n)|g(n) in (5.3), where g is any multiplicative function which guarantees that f is multiplicative, we can simplify af,s(p). We have that |µ(pm)| is 0 for m ≥ 2 and 1 for m = 1, since |µ(n)| is the characteristic function of the squarefree integers. If , we also have that , which means that P∞ f(pm) g(p) . m = 1 f(p) = |µ(p)|g(p) = g(p) af,s(p) = m=1 pms = ps We still retain the g(p) condition. Assuming that the sum over primes converges, substituting into af,s(p) = ps 6= −1 (5.3) gives ! g(p) ! X |µ(n)|ω(n)g(n) X |µ(n)|g(n) X ps = . ns ns 1 + g(p) n∈N n∈N p ps We utilize the same convergence criterion as Theorem 14. Simplifying the fraction and letting g be represented by f in order to maintain consistent notation completes the proof. Theorem 17. Let f(n) be a multiplicative function. Let P µ(n)f(n) and P f(p) both converge absolutely n∈N ns p ps−f(p) for . Assume that for all prime and such that we have s. Then σ > σa p s ∈ C σ > σa f(p) 6= p ! ! X µ(n)ω(n)f(n) X µ(n)f(n) X f(p) (5.6) = , ns ns f(p) − ps n∈N n∈N p which converges absolutely for σ > σa. Proof. We let f(n) = µ(n)g(n) in (5.3), where g is any multiplicative function which guarantees that f is multiplica- tive. We can then simplify , since P∞ f(pm) g(p) . We still retain the g(p) af,s(p) af,s(p) := m=1 pms = − ps af,s(p) = − ps 6= −1 condition. Assuming that the sum over primes converges, substituting into (5.3) gives ! g(p) ! X µ(n)ω(n)g(n) X µ(n)g(n) X − ps = . ns ns 1 − g(p) n∈N n∈N p ps We utilize the same convergence criterion as Theorem 14. Simplifying the fraction and letting g be represented by f in order to maintain consistent notation completes the proof. We note that the following proposition, the simplest application of (5.3), can be found in [4, (D-17)]. Corollary 18. Let ζ(s) be the Riemann zeta function. For s ∈ C such that σ > 1 , X ω(n) (5.7) = ζ(s)P (s). ns n∈N Proof. We let in (5.4), since this is a completely multiplicative function. We then note that P 1 is the f(n) = 1 p ps prime zeta function. The zeta and prime zeta functions both converge absolutely for σ > 1, so the left-hand side will too. Corollary 19. Let λ(s) be Liouville's function. For s ∈ C such that σ > 1, X ω(n)λ(n) ζ(2s) (5.8) = − P (s). ns ζ(s) n∈N SUMS INVOLVING THE NUMBER OF DISTINCT PRIME FACTORS FUNCTION Proof. We let f(n) = λ(n) in (5.4). Liouville's function, found in [8, (27.2.13)], is completely multiplicative. We note that λ(p) = −1 for every prime p, since they trivially only have a single prime divisor with multiplicity 1. This gives ! ! X ω(n)λ(n) X λ(n) X (−1) = . ns ns ps n∈N n∈N p We then note that [8, (27.4.7)] gives P λ(n) = ζ(2s) and states that it converges for σ > 1, which completes the n∈N ns ζ(s) proof since both the zeta and prime zeta functions converge for σ > 1. Corollary 20. Let χ(n) denote a Dirichlet character. For s ∈ C such that σ > 1 , X ω(n)χ(n) X χ(p) (5.9) = L(s, χ) . ns ps n∈N p Proof. We let f(n) = χ(n) in (5.4). Here χ(n) is a Dirichlet character, found in [8, (27.8.1)], which is a completely multiplicative function that is periodic with period k and vanishes for (n, k) > 1. Substituting it in (5.4) gives ! ! X ω(n)χ(n) X χ(n) X χ(p) = . ns ns ps n∈N n∈N p We then note that a Dirichlet L-series, an important number theoretic series, is de ned in [8, (25.15.1)] as L(s, χ) = P χ(n) . Simplifying to write the sum over n as an L-series while noting that [8, (25.15.1)] states that an L-series n∈N ns converges absolutely for σ > 1 completes the proof, since both the L series and prime L series converge absolutely for σ > 1. Corollary 21. For s ∈ C such that σ > 1 , X |µ(n)|ω(n) ζ(s) (5.10) = P (s, 1). ns ζ(2s) n∈N Proof. We let f(n) = 1 in (5.5) and note that [8, (27.4.8)] states that P |µ(n)| = ζ(s) and that this converges n∈N ns ζ(2s) for σ > 1. We note that (11) states that P (s, 1) will also converge absolutely for σ > 1, which completes the proof. Corollary 22. For s ∈ C such that σ > 1 , X µ(n)ω(n) 1 (5.11) = − P (s, −1). ns ζ(s) n∈N Proof. We let f(n) = 1 in (5.6) and note that [8, (27.4.5)] states P µ(n) = 1 and that it converges for σ > 1. n∈N ns ζ(s) We note that (11) states that P (s, −1) will also converge absolutely for σ > 1, which completes the proof. While the previous sums have involved completely multiplicative functions or convolutions with the Möbius function, we can sometimes directly evaluate P∞ f(pm) . Taking (5.3) but converting P f(n) back to its Euler m=1 pms n∈ ns product means that N X ω(n)f(n) Y X af,s(p) s = (1 + af,s(p)) , n 1 + af,s(p) n∈N p p where P∞ f(pm) . We can extract the coe cient through a variety of methods. af,s(p) := m=1 pms af,s(p) Corollary 23. For s ∈ C such that σ > 1, X ω(n)2ω(n) ζ2(s) (5.12) = 2 P (s, 1). ns ζ(2s) n∈N ω(n) 2 Proof. We know from [8, (27.4.9)] that P 2 = ζ (s) and that it converges for <(s) > 1, where 2ω(n) is the n∈N ns ζ(2s) number of squarefree divisors of n. We now directly evaluate af,s(p), summing it as a geometric series. We note ω(pm) m that 2 = 2, since p trivially has a single distinct prime factor. Substituting into the formula for af,s(p) shows that ∞ m ∞ m X 2ω(p ) X 1 2 a (p) = = 2 = . f,s pms ps ps − 1 m=1 m=1 SUMS INVOLVING THE NUMBER OF DISTINCT PRIME FACTORS FUNCTION af,s(p) 2 Then = s . Substituting into (5.3) shows that 1+af,s(p) p +1 ! ! X ω(n)2ω(n) X 2ω(n) X 2 = . ns ns ps + 1 n∈N n∈N p Rewriting the right-hand side in terms of zeta and prime zeta functions while noting that they will both converge if σ > 1 completes the proof. Corollary 24. Let denote Jordan's totient function. For such that , Jk(n) s, k ∈ C σ > max (1, 1 + <(k)) k X ω(n)Jk(n) ζ(s − k) X p − 1 (5.13) = . ns ζ(s) ps − 1 n∈N p k −1 k Proof. Taking p −1 , we have 1 1 and af,s(p) p −1 . Substituting af,s(p) = s k (1 + af,s(p)) = 1 − s 1 − s−k = s p −p p p 1+af,s(p) p −1 into (5.3) shows that −1 k ! X ω(n)Jk(n) Y 1 1 X p − 1 = 1 − 1 − . ns ps ps−k ps − 1 n∈N p p The series P pk−1 does not have a representation as a sum of prime zeta and shifted prime zeta functions in p ps−1 general, but in special cases such as s = 2k, k > 1, it does. Rewriting in terms of zeta functions and taking convergence criteria based on (12) completes the proof. We conclude with some additional nontrivial examples, which will perhaps be of use when considering exotic divisor sums. They are all special cases of Theorem 14 and can be proved similarly to Corollary 24. In each case we begin with a known Euler product for a function f(n), then use that to extract af,s(p). Corollary 25. For s, k ∈ C such that σ > max (1, 1 + <(k)), X ω(n)σk(n) (5.14) = ζ(s)ζ(s − k)(P (s) + P (s − k) − P (2s − k)) . ns n∈N Proof. We know from [8, (27.4.11)] that P σk(n) , where is the sum of the th powers s = ζ(s)ζ(s − k) σk(n) k n∈N n −1 −1 of the divisors of . Expanding the Euler product gives us that Q 1 1 n ζ(s)ζ(s − k) = p 1 − ps 1 − ps−k = −1 −1 Q . We obtain 1 1 . Simplifying gives that af,s(p) 1 1 (1 + af,s(p)) 1 + af,s(p) = 1 − s 1 − s−k = s + s−k − p p p 1+af,s(p) p p 1 . Substituting into (5.3) shows that p2s−k ! X ω(n)σk(n) X σk(n) X 1 1 1 = + − . ns ns ps ps−k p2s−k n∈N n∈N p Rewriting the right hand side in terms of zeta and prime zeta functions while noting that they will both converge if <(s) > max (1, 1 + <(k)) completes the proof. Corollary 26. Let s ∈ C. Then X ω(n)d(n2) ζ3(s) (5.15) = (4P (s, 1) − P (s)) , ns ζ(2s) n∈N which converges if <(s) > 1. Proof. We know from [7, (3.41)] that 2 X σa(n ) ζ(s)ζ(s − a)ζ(s − 2a) = . ns ζ(2s − 2a) n∈N As in the proof of (5.14), expanding the Euler product gives us that −1 −1 −1 ζ(s)ζ(s − a)ζ(s − 2a) Y 1 1 1 1 = 1 − 1 − 1 − 1 − . ζ(2s − 2a) ps ps−a ps−2a p2s−2a p SUMS INVOLVING THE NUMBER OF DISTINCT PRIME FACTORS FUNCTION The term inside the product is equal to (1 + af,s(p)) for every prime p, so on simplifying we get that −1 af,s(p) 1 1 1 1 2 1 1 1 = s + s−a + s−2a − 2s−a − 2s−2a − 2s−3a + 3s−3a 1 − 2s−2a . 1 + af,s(p) p p p p p p p p However, there is no convenient decomposition into a sum of terms of the form 1 and 1 so we restrict our pα pα±1 2 2 2 2 attention to the case a = 0. Then σ0(n ) = d(n ), where d(n ) counts the number of divisors of n . With a = 0, we have −1 af,s(p) 3 4 1 1 = s − 2s + 3s 1 − 2s . 1 + af,s(p) p p p p Using the identity p2s − 1 = (ps − 1)(ps + 1) allows us to simplify this to af,s(p) 4 1 = s − s . 1 + af,s(p) p + 1 p Substituting this into (5.3) gives us that ! X ω(n)d(n2) X d(n2) X 4 1 = − . ns ns ps + 1 ps n∈N n∈N p Rewriting in terms of zeta and prime zeta functions while nding convergence criteria to match both completes the proof. Corollary 27. Let s ∈ C. Then X ω(n)d2(n) ζ4(s) (5.16) = (8P (s, 1) + P (2s) − 4P (s)) , ns ζ(2s) n∈N which converges if <(s) > 1. Proof. Mirroring the proof of the last result, [7, (3.64)] shows that X σa(n)σb(n) ζ(s)ζ(s − a)ζ(s − b)ζ(s − a − b) = . ns ζ(2s − a − b) n∈N As in the proof of (5.14), expanding the Euler product gives us that −1 −1 −1 ζ(s)ζ(s − a)ζ(s − b)ζ(s − a − b) Y 1 1 1 = 1 − 1 − 1 − ζ(2s − a − b) ps ps−a ps−b p 1 −1 1 × 1 − 1 − . ps−a−b p2s−a−b The term inside the product is equal to (1 + af,s(p)) for every prime p, so on simplifying we get that af,s(p) 1 1 1 1 1 1 3 1 1 = s + s−a + s−b + s−a−b − 2s−a − 2s−b − 2s−a−b − 2s−a−2b − 2s−2a−b 1 + af,s(p) p p p p p p p p p 1 1 1 1 1 1 −1 + + + + − 1 − . p3s−a−b p3s−2a−b p3s−a−2b p3s−2a−2b p4s−2a−2b p2s−a−b However, there is no good decomposition into a sum of terms of the form 1 and 1 . Even the case proves pα pα±1 a = b 2 2 intractable so we restrict our attention to the case a = b = 0. Then σ0(n)σ0(n) = d (n), where d (n) is the square of the number of divisors of n. With a = b = 0, we have −1 af,s(p) 4 7 4 1 1 = s − 2s + 3s − 4s 1 − 2s . 1 + af,s(p) p p p p p Using the identity p2s − 1 = (ps − 1)(ps + 1) allows us to simplify this to af,s(p) 8 1 4 = s + 2s − s . 1 + af,s(p) p + 1 p p Substituting this into (5.3) gives us that ! X ω(n)d2(n) X d2(n) X 8 1 4 = + − . ns ns ps + 1 p2s ps n∈N n∈N p SUMS INVOLVING THE NUMBER OF DISTINCT PRIME FACTORS FUNCTION Rewriting in terms of zeta and prime zeta functions while nding convergence criteria to match both completes the proof. 6. Extensions Lastly, we show how to generalize the methods of this paper to second and higher order derivatives. We only sketch the following proofs, which can be made rigorous by paying su cient attention to convergence criteria. Proposition 28. We have the symmetric function identity n n ! n !2 n (6.1) X 2 k Y X xi X xi k ekt = (1 + xi) + 2 . 1 + xi k=1 i=1 i=1 i=1 (1 + xi) Proof. Starting with the generating product for , Qn Pn k, we study the action of the dierential ek k=1(1+txi) = k=0 ekt operator d . Applying it once yields D := t dt n ! n Y X k D (1 + txi) = kekt , k=1 k=1 from which we recover the familiar (3.1) after evaluating at t = 1. Applying D a second time results in n n n ! X 2 k Y X xi (6.2) k ekt = D t (1 + txi) 1 + txi k=1 k=1 k=1 2 n ! n ! n n 2 Y X xi X xi X x (6.3) = (1 + x ) + − i . i 1 + x 1 + x 2 i=1 i=1 i i=1 i i=1 (1 + xi) Some algebra completes the proof. In general, iterating the application of D will yield higher order identities for the elementary symmetric polyno- mials. Proposition 29. Let P∞ f(pm) as usual. If is multiplicative and has su ciently large real part af,s(p) := m=1 pms f s to ensure convergence of all the following series, then 2 ! !2 (6.4) X ω(n) f(n) X f(n) X af,s(p) X af,s(p) s = s + 2 . n n 1 + af,s(p) (1 + af,s(p)) n∈N n∈N p p If f is instead completely multiplicative, we have 2 ! !2 2 X ω(n) f(n) X f(n) X f(p) X f(p) X f(p) (6.5) = + − . ns ns ps ps ps n∈N n∈N p p p Proof. Letting xi = af,s(pi) in (6.1) while letting t → 1 and n → ∞ gives the rst result. Summing af,s(p) as a geometric series gives the second. As an example, letting f(n) = 1 gives 2 X ω(n) (6.6) = ζ(s) P 2(s) + P (s) − P (2s) , ns n∈N which converges for <(s) > 1. This result, though the simplest application of Proposition 29, appears to be new. k In general, applying D to E(t) k times will result in a Dirichlet series of the form P ω(n) f(n) . This may prove n∈N ns useful when considering asymptotics for the weighted k-th moments X ωk(n)f(n), n≤x for some multiplicative f. SUMS INVOLVING THE NUMBER OF DISTINCT PRIME FACTORS FUNCTION 7. Conclusion These theorems apply to any multiplicative functions. Together, this allows for a large class of in nite and divisor sums weighted by ω(n) to be addressed for the rst time. This leads to some surprising results such as that the Dirichlet series for products of ω(n) and other multiplicative functions often have a convenient closed form expression in terms of zeta and prime zeta functions. The methods of this paper also suggest an obvious generalization; taking the kth derivative of E(t) will lead to nite and in nite sums involving ω(n)k. This also suggests deep connections between the theory of symmetric functions and Dirichlet series, since with the right choice of xi we can interpret a Dirichlet series as a sum over symmetric polynomials. Dierent identities for symmetric polynomials with correspond to general expressions for Dirichlet series weighted by dierent functions. Due to this, a systematic study of identities for symmetric functions should correspond to surprising identities for Dirichlet series. Acknowledgements. I wrote this when I was 17, and looking back now I'm still amazed. This was my rst ever paper, and helped launch my dream of becoming a researcher 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 o 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, Josh Schuman, Jamie Andrews, and others who miraculously put up with me through the years. Go cougars. References [1] Apostol, T. M. Introduction to Analytic Number Theory. Springer-Verlag, New York, NY, 1976. [2] Erdös, P. and Kac, M.. The Gaussian law of errors in the theory of additive number theoretic functions. American Journal of Mathematics, 62:738742, 1940. [3] Fröberg, C.-E. On the prime zeta function. Nordisk Tidskr. Informationsbehandling (BIT), 8:187202, 1968. [4] Gould, H. W. and Shonhiwa, T.. A catalog of interesting Dirichlet series. Missouri J. Math. Sci., 1:218, 2008. [5] Hardy, G. H. and Ramanujan, S. The normal number of prime factors of a number n [Quart. J. Math. 48 (1917), 7692]. Collected papers of Srinivasa Ramanujan. AMS Chelsea Publ., Providence, RI, 2000. [6] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, New York, 1995. [7] Mathar, R. J. Survey of Dirichlet series of multiplicative arithmetic functions. Pre-print, 2012. http://arxiv.org/abs/1106.4038. [8] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov, Release 1.0.9 of 2014-08-29. Online companion to [9]. [9] Olver, F. W. J. and Lozier, D. W. and Boisvert, R. F. and Clark, C. W., editors. NIST Handbook of Mathematical Functions. Cambridge University Press, New York, NY, 2010. Print companion to [8]. ∗ Quince Orchard High School, Gaithersburg MD 20878 † Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg MD 20899 E-mail address: [email protected]