Abstract
For every positive integer n and for every \(\alpha \in [0, 1]\), let \({\mathcal {B}}(n, \alpha )\) denote the probabilistic model in which a random set \({\mathcal {A}} \subseteq \{1, \ldots , n\}\) is constructed by picking independently each element of \(\{1, \ldots , n\}\) with probability \(\alpha \). Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of \({\mathcal {A}}\).Let q be an indeterminate and let \([k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]\) be the q-analog of the positive integer k. We determine the expected value and the variance of \(X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )\), where \([{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}\). Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.
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1 Introduction
A nice consequence of the Prime Number Theorem is the asymptotic formula
where \({\text {lcm}}\) denotes the least common multiple. Indeed, precise estimates for \(\log {\text {lcm}}(1, \ldots , n)\) are equivalent to the Prime Number Theorem with an error term. Thus, a natural generalization is to study estimates for \(L_f(n) := \log {\text {lcm}}(f(1), \ldots , f(n))\), where f is a well-behaved function, for instance, a polynomial with integer coefficients. (We ignore terms equal to 0 in the \({\text {lcm}}\) and we set \({\text {lcm}}\varnothing := 1\).) When \(f \in {\mathbb {Z}}[x]\) 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 \(d \ge 3\), Cilleruelo [6] conjectured that \(L_f(n) \sim (d - 1)\, n \log n\), as \(n \rightarrow +\infty \), 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 \(\alpha \in [0, 1]\), let \({\mathcal {B}}(n, \alpha )\) denote the probabilistic model in which a random set \({\mathcal {A}} \subseteq \{1, \ldots , n\}\) is constructed by picking independently each element of \(\{1, \ldots , n\}\) with probability \(\alpha \). Cilleruelo et al. [9] studied the least common multiple of the elements of \({\mathcal {A}}\) 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).
Theorem 1.1
Let \({\mathcal {A}}\) be a random set in \({\mathcal {B}}(n, \alpha )\). Then, as \(\alpha n \rightarrow +\infty \), we have
with probability \(1 - o(1)\), where the factor involving \(\alpha \) is meant to be equal to 1 for \(\alpha = 1\).
Remark 1.1
In the deterministic case \(\alpha = 1\), we have \({\mathcal {A}} = \{1, \ldots , n\}\) (surely) and Theorem 1.1 corresponds to (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 \({\mathcal {S}}\) of positive integers, let \([{\mathcal {S}}]_q := \big \{[k]_q : k \in {\mathcal {S}}\big \}\).
The aim of this paper is to study the least common multiple of the elements of \([{\mathcal {A}}]_q\) for a random set \({\mathcal {A}}\) in \({\mathcal {B}}(n, \alpha )\). Our main results are the following:
Theorem 1.2
Let \({\mathcal {A}}\) be a random set in \({\mathcal {B}}(n, \alpha )\) and put \(X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )\). Then, for every integer \(n \ge 2\) and every \(\alpha \in [0,1]\), we have
where \({\text {Li}}_2(z) := \sum _{k=1}^\infty z^k / k^2\) is the dilogarithm and the factor involving \(\alpha \) is meant to be equal to 1 when \(\alpha = 1\). In particular,
as \(n \rightarrow +\infty \), uniformly for \(\alpha \in [0,1]\).
Theorem 1.3
Let \({\mathcal {A}}\) be a random set in \({\mathcal {B}}(n, \alpha )\) and put \(X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )\). Then there exists a function \(\mathrm {v} : {(0,1)} \rightarrow {\mathbb {R}}^+\) such that, as \(\alpha n / \big ((\log n)^3 (\log \log n)^2\big ) \rightarrow +\infty \), we have
Moreover, the upper bound
holds for every positive integer n and every \(\alpha \in [0, 1]\).
As a consequence of Theorems 1.2 and 1.3, we obtain the following q-analog of Theorem 1.1.
Theorem 1.4
Let \({\mathcal {A}}\) be a random set in \({\mathcal {B}}(n, \alpha )\). Then, as \(\alpha n \rightarrow +\infty \), we have
with probability \(1 - o(1)\), where the factor involving \(\alpha \) is meant to be equal to 1 for \(\alpha = 1\).
Remark 1.2
In the deterministic case \(\alpha = 1\), we have (see Lemma 4.1 below)
and Theorem 1.4 corresponds to the well-known asymptotic formula \(\sum _{d \le n} \varphi (d) \sim \tfrac{3}{\pi ^2} n^2\) (Lemma 3.3 below) for the sum of the first values of the Euler function \(\varphi \).
Remark 1.3
In Theorem 1.4 the condition \(\alpha n \rightarrow +\infty \) is necessary. Indeed, if \(\alpha n \le C\), for some constant \(C > 0\), then
as \(n \rightarrow +\infty \), and so no (nontrivial) asymptotic formula for \(\deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )\) 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 \(\alpha \in [0, 1]\) and for a random set \({\mathcal {A}}\) in \({\mathcal {B}}(n, \alpha )\), an appropriate normalization of the random variable \(\log {\text {lcm}}({\mathcal {A}})\) converges in distribution to a standard normal random variable, as \(n \rightarrow +\infty \). 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 \({\text {lcm}}\!\big ([f(1)]_q, \cdots , [f(n)]_q\big )\) for \(f \in {\mathbb {Z}}[x]\) or, more generally, \({\text {lcm}}\!\big ([f(k)]_q : k \in {\mathcal {A}}\big )\) with \({\mathcal {A}}\) a random set in \({\mathcal {B}}(n, \alpha )\).
2 Notation
We employ the Landau–Bachmann “Big Oh” and “little oh” notations O and o, as well as the associated Vinogradov symbol \(\ll \), 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 “\(X \sim Y\) with probability \(1 - o(1)\)”, as the parameters tend to some limit, if for every \(\varepsilon > 0\) we have \({\mathbb {P}}\big [\,|X - Y| > \varepsilon |Y|\,\big ] = o_\varepsilon (1)\), 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 \(\varphi (n)\), \(\mu (n)\), \(\tau (n)\), and \(\sigma (n)\), for the Euler totient function, the Möbius function, the number of divisors, and the sum of divisors, of a positive integer n, respectively.
3 Preliminaries
In this section we collect some preliminary results needed in later arguments.
Lemma 3.1
We have
for every \(x \ge 2\).
Proof
See, e.g., [20, Part I, Theorem 3.2]. \(\square \)
Lemma 3.2
We have
for every \(x \ge 2\).
Proof
From Lemma 3.1 and partial summation, it follows that
Let \(e := (e_1, e_2)\) and \(e_i^\prime := e_i / e\) for \(i=1,2\). Then we have
as desired. \(\square \)
Let us define
for every \(x \ge 1\) and for all positive integers \(a_1, a_2\).
Lemma 3.3
We have
for every \(x \ge 2\).
Proof
See, e.g., [20, Part I, Theorem 3.4]. \(\square \)
Lemma 3.4
We have
for every \(x \ge 2\), where
and the series is absolutely convergent.
Proof
From the identity \(\varphi (n) / n = \sum _{d \,\mid \;\;\!\!\! n} \mu (d) / d\), 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
is absolutely convergent. Therefore, we obtain
Now (5) follows from (7) by partial summation and since \(C_1(a_1, a_2) = \dfrac{a_1 a_2}{3}\,C_0(a_1, a_2)\). \(\square \)
Remark 3.1
The obvious bound \(\varphi (m) \le m\) yields \(C_1(a_1, a_2) \le \dfrac{a_1 a_2}{3}\) (which is not so obvious from (6)).
We end this section with an easy observation that will be useful later.
Remark 3.2
It holds \(1 - (1 - x)^k \le k x\), for all \(x \in [0, 1]\) and for all integers \(k \ge 0\).
4 Proofs
Henceforth, let \({\mathcal {A}}\) be a random set in \({\mathcal {B}}(n, \alpha )\), let \([{\mathcal {A}}]_q\) be its q-analog, and put \(L := {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )\) and \(X := \deg L\). For every positive integer d, let us define
The following lemma gives a formula for X in terms of \(I_{{\mathcal {A}}}\) and the Euler function.
Lemma 4.1
We have
Proof
For every positive integer k, it holds
where \(\Phi _d(q)\) is the dth cyclotomic polynomials. Since, as it is well known, every cyclotomic polynomial is irreducible over \({\mathbb {Q}}\), it follows that L is the product of the polynomials \(\Phi _d(q)\) such that \(d > 1\) and \(d \mid k\) for some \(k \in {\mathcal {A}}\). Finally, the equality \(\deg \!\big (\Phi _d(q)\big ) = \varphi (d)\) and the definition of \(I_{{\mathcal {A}}}\) yield (8). \(\square \)
Let \(\beta := 1 - \alpha \). The next lemma provides two expected values involving \(I_{{\mathcal {A}}}\).
Lemma 4.2
For all positive integers \(d, d_1, d_2\), we have
and
Proof
On the one hand, by the definition of \(I_{{\mathcal {A}}}\), 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. \(\square \)
We are ready to compute the expected value of X.
Proof of Theorem 1.2
From Lemmas 4.1 and 4.2, it follows that
Moreover, since \(\lfloor n / d \rfloor = j\) if and only if \(n / (j + 1) < d \le n / j\), 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 \(1 - \beta ^n = O(\alpha n)\), we get (2). The proof is complete. \(\square \)
Now we consider the variance of X.
Proof of Theorem 1.3
From Lemmas 4.1 and 4.2, it follows that
Let us define
Clearly, we have
Hence, in order to prove (3), it suffices to show that \(V_n(\alpha ) = \mathrm {v}(\alpha ) + o(1)\).
For all vectors \(\varvec{a} := (a_1, a_2)\) and \(\varvec{j} := (j_1, j_2, j_3)\) with components in the set of positive integers, define the quantities
and
Let \(d := (d_1, d_2)\) and \(a_i := d_i / d\) for \(i=1,2\). Then the equalities
are equivalent to
which in turn are equivalent to
that is,
Therefore, letting
and
we have
Now let us define
where
and
recalling that \(C_1(a_1, a_2)\) is defined by (6). The convergence of series (13) follows easily from Remark 3.1, \(\rho _2(\varvec{a}, \varvec{j}) \le 1 / (a_1 a_2 j_3)\), and the fact that \(\min (j_1, j_2) \ge j_3\) for all \((\varvec{a}, \varvec{j}) \in {\mathcal {S}}_\infty \).
Thanks to Lemma 3.4, for each \((\varvec{a}, \varvec{j}) \in {\mathcal {S}}_n\) we have
Consequently, we get that
where
and
Now we have to bound both \(\Sigma _1\) and \(\Sigma _2\).
If \((\varvec{a}, \varvec{j}) \in {\mathcal {S}}_\infty \setminus {\mathcal {S}}_n\) then \(\big (\rho _2(\varvec{a}, \varvec{j}) - \rho _1(\varvec{a}, \varvec{j})\big ) n < 1\) and consequently, also by Remark 3.1,
where, for brevity, we wrote \(\rho _i := \rho _i(\varvec{a}, \varvec{j})\) for \(i=1,2\).
If \((\varvec{a}, \varvec{j}) \in {\mathcal {S}}_\infty \) then, as we already noticed, \(\min (j_1, j_2) \ge j_3\) and, moreover,
Hence, we have
where we used the inequality \(1 - \beta ^j \le \alpha j\), which follows from Remark 3.2.
On the one hand, from (15) and (16) it follows that
as \(\alpha n / \!\big ((\log n)^3 (\log \log n)^2\big ) \rightarrow +\infty \) (actually, \(\alpha n / \!\log n \rightarrow +\infty \) is sufficient).
On the other hand, from \(\rho _2(\varvec{a}, \varvec{j}) \le 1 / (a_1 a_2 j_3)\), (16), and the bound \(\sigma (m) \ll m \log \log m\) (see, e.g., [20, Part I, Theorem 5.7]) it follows that
as \(\alpha n / \big ((\log n)^3 (\log \log n)^2\big ) \rightarrow +\infty \).
At this point, putting together (14), (17), and (18), we obtain \(V_n(\alpha ) = \mathrm {v}(\alpha ) + o(1)\), as desired. The proof of (3) is complete.
It remains only to prove the upper bound (4). From (12) it follows that
where we used Remark 3.2, Lemma 3.1, and the bound \(n! > (n / \mathrm {e})^n\). Thus (4) is proved. \(\square \)
Proof of Theorem 1.4
By Chebyshev’s inequality, Theorems 1.2 and 1.3, we have
as \(\alpha n \rightarrow +\infty \). Hence, using again Theorem 1.2, we get
with probability \(1 - o(1)\), as \(\alpha n \rightarrow +\infty \). \(\square \)
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C. Sanna is a member of GNSAGA of INdAM and of CrypTO, the group of Cryptography and Number Theory of Politecnico di Torino
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Sanna, C. On the least common multiple of random q-integers. Res. number theory 7, 16 (2021). https://doi.org/10.1007/s40993-021-00242-4
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DOI: https://doi.org/10.1007/s40993-021-00242-4