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Completely Multiplicative Functions Taking Values in {−1, 1}

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Completely Multiplicative Functions Taking Values in {−1, 1} TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 12, December 2010, Pages 6279–6291 S 0002-9947(2010)05235-3 Article electronically published on July 14, 2010 COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN {−1, 1} PETER BORWEIN, STEPHEN K. K. CHOI, AND MICHAEL COONS Abstract. Define the Liouville function for A, a subset of the primes P ,by Ω (n) λA(n)=(−1) A ,whereΩA(n) is the number of prime factors of n coming from A counting multiplicity. For the traditional Liouville function, A is the set of all primes. Denote LA(n) LA(x):= λA(n)andRA := lim . n→∞ n n≤x It is known that for each α ∈ [0, 1] there is an A ⊂ P such that RA = α. Given certain restrictions on the sifting density of A, asymptotic estimates for n≤x λA(n) can be given. With further restrictions, more can be said. For an odd prime p, define the character–like function λp as λp(pk + i)=(i/p) for i =1,...,p− 1andk ≥ 0, and λp(p)=1,where(i/p) is the Legendre symbol (for example, λ3 is defined by λ3(3k +1)= 1, λ3(3k +2)= −1(k ≥ 0) and λ3(3) = 1). For the partial sums of character–like functions we give exact values and asymptotics; in particular, we prove the following theorem. Theorem. If p is an odd prime, then max λp(k) log x. n≤x k≤n This result is related to a question of Erd˝os concerning the existence of bounds for number–theoretic functions. Within the course of discussion, the ratio φ(n)/σ(n) is considered. 1. Introduction Let Ω(n) be the number of distinct prime factors in n (with multiple factors counted multiple times). The Liouville λ–function is defined by λ(n):=(−1)Ω(n). Therefore, λ(1) = λ(4) = λ(6) = λ(9) = λ(10) = 1 and λ(2) = λ(5) = λ(7) = λ(8) = −1. In particular, λ(p)=−1 for any prime p. Itiswellknown(e.g.see §22.10 of [10]) that Ω is completely additive; i.e, Ω(mn)=Ω(m)+Ω(n) for any m and n, and hence λ is completely multiplicative, i.e., λ(mn)=λ(m)λ(n) for all m, n ∈ N. It is interesting to note that on the set of square-free positive integers λ(n)=μ(n), where μ is the M¨obius function. In this respect, the Liouville λ– function can be thought of as an extension of the M¨obius function. Received by the editors June 13, 2008. 2000 Mathematics Subject Classification. Primary 11N25, 11N37; Secondary 11A15. Key words and phrases. Liouville lambda function, multiplicative functions. This research was supported in part by grants from NSERC of Canada and MITACS. c 2010 by the authors 6279 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6280 PETER BORWEIN, STEPHEN K. K. CHOI, AND MICHAEL COONS Similar to the M¨obius function, many investigations surrounding the λ–function concern the summatory function of initial values of λ; that is, the sum L(x):= λ(n). n≤x Historically, this function has been studied by many mathematicians, including Liouville, Landau, P´olya, and Tur´an. Recent attention to the summatory function of the M¨obius function has been given by Ng [16, 17]. Larger classes of completely multiplicative functions have been studied by Granville and Soundararajan [7, 8, 9]. One of the most important questions is that of the asymptotic order of L(x). More formally, the question is to determine the smallest value of ϑ for which L(x) lim =0. x→∞ xϑ It is known that the value of ϑ = 1 is equivalent to the prime number theorem 1 [14, 15] and that ϑ = 2 + ε for any arbitrarily small positive constant ε is equiv- 1 alent to the Riemann√ hypothesis [3]. (The value of 2 + ε is best possible, as lim supx→∞ L(x)/ x>.061867; see Borwein, Ferguson, and Mossinghoff [4].) In- ∈ 1 deed, any result asserting a fixed ϑ 2 , 1 would give an expansion of the zero-free region of the Riemann zeta function, ζ(s), to (s) ≥ ϑ. Unfortunately, a closed form for determining L(x) is unknown. This brings us to the motivating question behind this investigation: are there functions similar to λ so that the corresponding summatory function does yield a closed form? Throughout this investigation P will denote the set of all primes. As an analogue to the traditional λ and Ω, define the Liouville function for A ⊂ P by ΩA(n) λA(n)=(−1) , where ΩA(n) is the number of prime factors of n coming from A counting multi- plicity. Alternatively, one can define λA as the completely multiplicative function with λA(p)=−1 for each prime p ∈ A and λA(p) = 1 for all p/∈ A.Everycom- pletely multiplicative function taking only ±1 values is built this way. The class of functions from N to {−1, 1} is denoted F({−1, 1}) (as in [8]). Also, define LA(n) LA := λA(n)andRA := lim . n→∞ n n≤x In this paper, we first consider questions regarding the properties of the function λA by studying the function RA. The rest of this paper considers an extended investigation of those functions in F({−1, 1}) which are character–like in nature (meaning that they agree with a real Dirichlet character χ at non-zero values). While these functions are not really direct analogues of λ, much can be said about them. Indeed, we can give exact formulae and sharp bounds for their partial sums. Within the course of discussion, the ratio φ(n)/σ(n) is considered. 2. Properties of LA(x) Define the generalized Liouville sequence as LA := {λA(1),λA(2),...}. Theorem 1. The sequence LA is not eventually periodic. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN {−1, 1} 6281 Proof. Towards a contradiction, suppose that LA is eventually periodic; say the sequence is periodic after the M–th term and has period k.NowthereisanN ∈ N such that for all n ≥ N,wehavenk > M.SinceA = ∅, pick p ∈ A.Then λA(pnk)=λA(p) · λA(nk)=−λA(nk). But pnk ≡ nk(mod k), a contradiction to the eventual k–periodicity of LA. Corollary 1. If A ⊂ P is non-empty, then λA is not a Dirichlet character. Proof. This is a direct consequence of the non–periodicity of LA. To get more acquainted with the sequence LA, we study the partial sums LA(x) of LA, and to study these, we consider the Dirichlet series with coefficients λA(n). Starting with singleton sets {p} of the primes, a nice relation becomes apparent; for (s) > 1 − ∞ (1 − p s) λ{ }(n) (1) ζ(s)= p , (1 + p−s) ns n=1 and for sets {p, q}, − − ∞ (1 − p s)(1 − q s) λ{ }(n) (2) ζ(s)= p,q . (1 + p−s)(1 + q−s) ns n=1 For any subset A of primes, since λA is completely multiplicative, for (s) > 1 we have ∞ ∞ λ (n) λ (pl) L (s):= A = A A ns pls n=1 p l=0 ∞ ∞ (−1)l 1 1 1 = = ls ls 1 − 1 ∈ p ∈ p ∈ 1+ ps ∈ 1 ps p A l=0 pA l=0 p A p A 1 − p−s (3) = ζ(s) . 1+p−s p∈A This relation leads us to our next theorem, but first let us recall a vital piece of notation from the Introduction. Definition 1. For A ⊂ P denote λA(1) + λA(2) + ...+ λA(n) RA := lim . n→∞ n The existence of the limit RA is guaranteed by Wirsing’s Theorem. In fact, Wirsing in [20] showed more generally that every real multiplicative function f with |f(n)|≤1 has a mean value; i.e., the limit 1 lim f(n) x→∞ x n≤x exists. Furthermore, in [19] Wintner showed that 1 f(p) f(p2) 1 lim f(n)= 1+ + + ... 1 − =0 x→∞ x p p2 p ≤ p n x | − | if and only if p 1 f(p) /p converges; otherwise the mean value is zero. This gives the following theorem. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 6282 PETER BORWEIN, STEPHEN K. K. CHOI, AND MICHAEL COONS Theorem 2. For the completely multiplicative function λA(n),thelimitRA exists and p−1 −1 ∞ p∈A p+1 if p∈A p < , (4) RA = 0 otherwise. p−1 Example 1. For any prime p, R{p} = p+1 . To be a little more descriptive, let us make some notational comments. Denote by P(P ) the power set of the set of primes. Note that p − 1 2 =1− . p +1 p +1 Recall from above that R : P(P ) → R is defined by 2 R := 1 − . A p +1 p∈A It is immediate that R is bounded above by 1 and below by 0, so that we need only consider that R : P(P ) → [0, 1].ItisalsoimmediatethatR∅ =1andRP =0. Remark 1. For an example of a subset of primes with mean value in (0, 1), consider the set K of primes defined by K := pn ∈ P : pn =min{q ∈ P } for n ∈ N . q>n3 Since there is always a prime in the interval (x, x + x5/8] (see Ingham [12]), these primes are well defined; that is, pn+1 >pn for all n ∈ N. The first few values give K = {11, 29, 67, 127, 223, 347, 521, 733, 1009, 1361,...}.
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