A WEIGHTED VERSION OF THE ERDOS-KAC} THEOREM RIZWANUR KHAN, MICAH B. MILINOVICH, AND UNIQUE SUBEDI Abstract. Let !(n) denote the number of distinct prime factors of n and let τk(n) denote the k-fold divisor function. We evaluate the centralized moments of !(n), weighted by τk(n), and deduce a weighted version of the Erd}os-KacTheorem. 1. Introduction Let !(n) denote the number of distinct prime factors of natural number n. That is, X !(n) = 1: pjn The celebrated Erd}os{Kactheorem [6] from 1940 states that for each α 2 R, we have Z α 1 X 1 2 (1.1) 1 −! p e−t =2 dt x 2π −∞ n≤x p !(n)−log log x≤α log log x as x ! 1. In other words, since the log log function grows very slowly, the quantity !(n) − log log n (1.2) p log log n follows a Gaussian distribution law with mean 0 and variancep 1. R´enyi and Tur´an[17] provided a (best possible) quantitative version of the theorem with O(1= log log x) as the rate of convergence in (1.1). Although the Erd}os{Kac theorem is a classical result, it has remained a source of great research interest even in modern times. Many notable mathematicians have revisited the theorem and pro- vided different proofs of it, e.g. [6, 14, 17, 10, 1, 9, 13, 11]. We highlight a couple of these approaches in particular. Generalizing the prime counting function π(x), let πk(x) denote the number of integers n ≤ x with !(n) = k. One approach to the Erd}os{Kactheorem, related to the proof of R´enyi and Tur´an[17] and thep Selberg{Delange Method [16, Chapter 7.4], is to use asymptotics for πk(x) with k ≤ log log x + α log log x to evaluate the left hand side of (1.1). Such a proof is rather involved, requiring complex analysis and the theory of the Riemann zeta function, and it is at least as deep as the prime number theorem. Another approach is to asymptotically evaluate the moments of the quantity (1.2) and show that they match the moments of a standard Gaussian random variable. Since a Gaussian distribution is completely characterized by its moments [2, Theorem 30.1], this implies the Erd}os{KacTheorem. The moments approach was first accomplished by Halberstam [10] in 1955, although his proof was rather complicated. About a decade later, Billingsley [1] gave a much simpler demonstration. In 2007, Granville and Soundararajan [9] provided an even more transparent and flexible treatment. Their approach remains one of the most direct and elementary ways to prove the Erd}os-Kactheorem. 2020 Mathematics Subject Classification. 11N60. Key words and phrases. Erd}os{Kactheorem, prime omega function, divisor function, central limit theorem, moments. 1 2 RIZWANUR KHAN, MICAH B. MILINOVICH, AND UNIQUE SUBEDI This paper is concerned with a generalization of the Erd}os-Kactheorem in which the distribution of !(n) is studied, but counting is weighted by the divisor function. Elliott [4, 5] proved that1 −1 Z α X X 1 2 (1.3) τ(n) τ(n) −! p e−t =2 dt; 2π −∞ n≤x n≤x p !(n)−2 log log x≤α 2 log log x P where τ(n) = djn 1 is the divisor function. Thus, when weighted by the divisor function, !(n) still follows a Gaussian distribution as in the Erd}os{Kactheorem, but with double the mean and double the variance. Such a result can be predicted by the following heuristic. Recall that τ(n) = 2!(n) for square-free n. So, roughly speaking, we are studying the Gaussian distributed !(n), tilted by its exponential. Consider a Gaussian random variable with mean 0 and variance 1, so that its distribution function is α 1 Z −x2 p e 2 dx: 2π −∞ If we weight the measure by ex, the distribution function becomes p Z α 2 Z α 2 1 −x x e −(x−1) p e 2 e dx = p e 2 dx; 2π −∞ 2π −∞ where equality is obtained by completing the square. Thus, the resulting distribution with the weighted measure is still Gaussian but with altered mean and variance. In Probability theory, this phenomenon is a simple case of Girsanov's Theorem [12, Chapter 3.5]. We also mention that Elliott's P result in (1.3) was generalized to short interval sums x≤n≤x+y, where y is a small power of x, by Liu and Wu [15], using similar methods. Weighted central limit theorems are also of current interest in other parts of number theory. For 1 instance, there is a famous and classical result of Selberg which establishes that log jζ( 2 + it)j has a Gaussian distribution for t 2 [T; 2T ] as T ! 1, where ζ(s) denotes the Riemann zeta-function. Recently, Fazzari [7, 8] has proved weighted versions of Selberg's central limit theorem assuming the Riemann hypothesis. Our paper, which was mostly written before we were aware of Elliot's previous work, was inspired by Fazzari's results for the Riemann zeta-function. Elliott's proof of (1.3) is essentially based on a Selberg{Delange type approach to the Erd}os{Kac theorem, and thus is quite deep. The goal of this paper is to give a relatively simple proof of Elliott's result using a moments approach. Having a moments result to this weighted problem is natural, given the historical development of the Erd}os{Kactheorem. This was lacking in the literature prior to our work. Also, while Elliott's weighting was by the divisor function, we generalize this result to weighting by the k-fold divisor function, τk(n). We compute the centralized moments of !(n) weighted by τk(n) by generalizing a method of Granville and Soundararajan [9]. Although the foundation of our approach was laid out elegantly in [9], our proof requires a number of nontrivial modifications. It was not clear, a priori, whether the Granville-Soundararajan approach would work in this case and our generalization requires a careful set-up. We now state our main theorem. Let X τk(n) = 1 n1···nk=n be the k-fold divisor function. In Section 2.2, we show that, as n ranges over the integers below x, the mean of !(n) with respect to the weighted measure τk(n) is asymptotic to k log log x. Thus we centralize by k log log x and evaluate the following m-th moment for (!(n) − k log log x), weighted by τk(n). 1Elliott more generally proved (1.3) with τ(n)α for any α > 0. A WEIGHTED VERSION OF THE ERDOS-KAC} THEOREM 3 Theorem 1.1. Let k and m be fixed natural numbers. For x > 3, we have X m !(n) − k log log x τk(n) n≤x X τk(n) (1.4) n≤x ( m=2 m−1 (m − 1)!! k log log x + O (log log x) 2 ; if m is even; = m−1 O (log log x) 2 ; if m is odd; where (m − 1)!! denotes the product of all odd integers up to and including (m − 1). Dividing both sides by (k log log x)m=2, this gives that the weighted m-th moment of !(n) − k log log x p k log log x is (m − 1)!! + o(1) if m is even and is o(1) if m is odd. Recall that the m-th moment of a standard Gaussian random varriable is (m − 1)!! if m is even and is 0 if m is odd. Thus, Theorem 1.1 implies the following weighted version of the Erd}os{KacTheorem. Corollary 1.2. Fix k; m 2 N. For any α 2 R; we have −1 Z α X X 1 −t2=2 (1.5) τk(n) τk(n) −! p e dt 2π −∞ n≤x n≤x p !(n)−k log log x≤α k log log x as x ! 1. Throughout this paper, we will follow the "-convention, where " will always denote an arbitrarily small positive constant, but not necessarily the same one from one occurrence to the next. Our error terms are allowed to depend on ". 2. Preliminaries 2.1. Averages of the k-fold divisor function. A well known result, due to Voronoi and Landau (see [19, Theorem 12.2]), states that for s X x k k−1 +" (2.1) τk(n) = Res ζ (s) + O x k+1 s=1 s n≤x for x ≥ 1 and k ≥ 2. The leading order term of the residue is x (log x)k−1 ; (k − 1)! while the lower order terms are proportional to x(log x)k−1−c for integers c with 1 ≤ c ≤ k − 1. We will need to sum τk(n) with n divisible by a fixed a 2 N. We prove such a result with a weaker error term than (2.1). This suffices for our application, as we only require a power savings in x=a below. Lemma 2.1. For a 2 N and x ≥ a, we have s k+3 X x k x k+6 +" !(a) (2.2) τk(n) = Res ζ (s)F (s; a) + O τk(a) M ; s=1 s a n≤x ajn where υ −1 k p m Y 1 X τk(p ) (2.3) F (s; a) := 1 − 1 − ps pms pυp jja m=0 4 RIZWANUR KHAN, MICAH B. MILINOVICH, AND UNIQUE SUBEDI and p 2 + 16 (2.4) M := p : 2 − 1 Though the function F (s; a) depends on k, we suppress this in the notation since k is assumed to be fixed.