On the least common multiple of random q-integers

For every positive integer n and for every α∈[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in [0, 1]$$\end{document}, let B(n,α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}(n, \alpha )$$\end{document} denote the probabilistic model in which a random set A⊆{1,…,n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}} \subseteq \{1, \ldots , n\}$$\end{document} is constructed by picking independently each element of {1,…,n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1, \ldots , n\}$$\end{document} with probability α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document}.Let q be an indeterminate and let [k]q:=1+q+q2+⋯+qk-1∈Z[q]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]$$\end{document} be the q-analog of the positive integer k. We determine the expected value and the variance of X:=deglcm([A]q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )$$\end{document}, where [A]q:={[k]q:k∈A}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}$$\end{document}. Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.


Introduction
A nice consequence of the Prime Number Theorem is the asymptotic formula log lcm(1, 2, . . . , n) ∼ n, as n → +∞, where lcm denotes the least common multiple. Indeed, precise estimates for log lcm (1, . . . , n) are equivalent to the Prime Number Theorem with an error term. Thus, a natural generalization is to study estimates for L f (n) := log lcm(f (1), . . . , f (n)), where f is a wellbehaved function, for instance, a polynomial with integer coefficients. (We ignore terms equal to 0 in the lcm and we set lcm ∅ := 1.) When f ∈ Z[x] is a linear polynomial, the product of linear polynomials, or an irreducible quadratic polynomial, asymptotic formulas for L f (n) were proved by Bateman et al. [3], Hong et al. [10], and Cilleruelo [6], respectively. In particular, for f (x) = x 2 + 1, Rué et al. [15] determined a precise error term for the asymptotic formula. When f is an irreducible polynomial of degree d ≥ 3, Cilleruelo [6] conjectured that L f (n) ∼ (d − 1) n log n, as n → +∞, but this is still an open problem. However, bounds for L f (n) were proved by Maynard and Rudnick [13], and Sah [16]. Moreover, Rudnick and Zehavi [14] studied the growth of L f (n) along a shifted family of polynomials. Another direction of research consists in considering the least common multiple of random sets of positive integers. For every positive integer n and every α ∈ [0, 1], let B(n, α) denote the probabilistic model in which a random set A ⊆ {1, . . . , n} is constructed by picking independently each element of {1, . . . , n} with probability α. Cilleruelo et al. [9] studied the least common multiple of the elements of A and proved the following result (see [1] for a more precise version, and [4,5,7,8,12,[17][18][19] for other results of a similar flavor). A be a random set in B(n, α). Then, as αn → +∞, we have

Theorem 1.1 Let
with probability 1 − o(1), where the factor involving α is meant to be equal to 1 for α = 1. Let q be an indeterminate. The q-analog of a positive integer k is defined by The q-analogs of many other mathematical objects (factorial, binomial coefficients, hypergeometric series, derivative, integral...) have been extensively studied, especially in Analysis and Combinatorics [2,11]. For every set S of positive integers, let [S] q := [k] q : k ∈ S . The aim of this paper is to study the least common multiple of the elements of [A] q for a random set A in B(n, α). Our main results are the following: Theorem 1.2 Let A be a random set in B(n, α) and put X := deg lcm [A] q . Then, for every integer n ≥ 2 and every α ∈ [0, 1], we have where Li 2 (z) := ∞ k=1 z k /k 2 is the dilogarithm and the factor involving α is meant to be equal to 1 when α = 1. In particular, as n → +∞, uniformly for α ∈ [0, 1].

Theorem 1.3 Let
A be a random set in B(n, α) and put X := deg lcm [A] q . Then there exists a function v : (0, 1) → R + such that, as αn/ (log n) 3 (log log n) 2 → +∞, we have Moreover, the upper bound holds for every positive integer n and every α ∈ [0, 1].
As a consequence of Theorems 1.2 and 1.3, we obtain the following q-analog of Theorem 1.1.

Theorem 1.4 Let
A be a random set in B(n, α). Then, as αn → +∞, we have as n → +∞, and so no (nontrivial) asymptotic formula for deg lcm [A] q can hold with probability 1 − o(1).
We conclude this section with some possible questions for further research on this topic. Alsmeyer, Kabluchko, and Marynych [1, Corollary 1.5] proved that, for fixed α ∈ [0, 1] and for a random set A in B(n, α), an appropriate normalization of the random variable log lcm(A) converges in distribution to a standard normal random variable, as n → +∞. In light of Theorems 1.2 and 1.3, it is then natural to ask whether the random variable converges in distribution to a normal random variable, or to some other random variable. Another problem could be considering polynomial values, similarly to the results done in the context of integers, and studying lcm [f (1)

Notation
We employ the Landau-Bachmann "Big Oh" and "little oh" notations O and o, as well as the associated Vinogradov symbol , with their usual meanings. Any dependence of the implied constants is explicitly stated or indicated with subscripts. For real random variables X and Y , depending on some parameters, we say that "X ∼ Y with probability 1 − o(1)", as the parameters tend to some limit, if for every ε > 0 we have P |X − Y | > ε|Y | = o ε (1), as the parameters tend to the limit. We let (a, b) and [a, b] denote the greatest common divisor and the least common multiple, respectively, of two integers a and b. As usual, we write ϕ(n), μ(n), τ (n), and σ (n), for the Euler totient function, the Möbius function, the number of divisors, and the sum of divisors, of a positive integer n, respectively.

Preliminaries
In this section we collect some preliminary results needed in later arguments.
Let e := (e 1 , e 2 ) and e i := e i /e for i = 1, 2. Then we have as desired.
Let us define for every x ≥ 1 and for all positive integers a 1 , a 2 .

Lemma 3.3 We have
for every x ≥ 2.

Lemma 3.4 We have
for every x ≥ 2, where C 1 (a 1 , a 2 ) := a 1 a 2 3 and the series is absolutely convergent.
Proof From the identity ϕ(n)/n = d |n μ(d)/d, it follows that Let c i := (a i , d i ) and e i := d i /c i , for i = 1, 2. On the one hand, we have On the other hand, thanks to Lemma 3.2, we have which, in particular, implies that the series C 0 (a 1 , a 2 ) := is absolutely convergent. Therefore, we obtain Now (5) follows from (7) by partial summation and since C 1 (a 1 , a 2 ) = a 1 a 2 3 C 0 (a 1 , a 2 ).
We end this section with an easy observation that will be useful later.  The following lemma gives a formula for X in terms of I A and the Euler function.

Lemma 4.1
We have Proof For every positive integer k, it holds where d (q) is the dth cyclotomic polynomials. Since, as it is well known, every cyclotomic polynomial is irreducible over Q, it follows that L is the product of the polynomials d (q) such that d > 1 and d | k for some k ∈ A. Finally, the equality deg d (q) = ϕ(d) and the definition of I A yield (8).
Let β := 1 − α. The next lemma provides two expected values involving I A .

Lemma 4.2 For all positive integers d, d 1 , d 2 , we have
and Proof On the one hand, by the definition of I A , we have which is (9). On the other hand, by linearity of the expectation and by (9), we have where the last expected value can be computed as and second claim follows.
We are ready to compute the expected value of X.
Moreover, since n/d = j if and only if n/(j + 1) < d ≤ n/j, we get that where we used Lemma 3.3. Putting together (10) and (11), and noting that, by Remark 3.2, the addend of (11) corresponding to d = 1 is 1 − β n = O(αn), we get (2). The proof is complete.
Now we consider the variance of X.

Authors' contributions
The author thanks the anonymous referee, whose careful reading and detailed suggestions led to a considerable improvement of the paper.