On the error term of the logarithm of the lcm of a quadratic sequence

Rué, Juanjo; Šarka, Paulius; Zumalacárregui, Ana

doi:10.5802/jtnb.843

Cited by 7 publications

(6 citation statements)

References 8 publications

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“…A similar growth occurs for products of linear polynomials, see [4], and for any polynomial with non-negative integer coefficients, there is a lower bound log L f (N ) ≫ N [3]. However, in the case of irreducible polynomials higher degree, Cilleruelo [2] conjectured that the growth is faster than linear, precisely: see also [7]. No other case of Conjecture 1.1 is known to date.…”

mentioning

confidence: 79%

On Cilleruelo's conjecture for the least common multiple of polynomial sequences

Rudnick¹,

Zehavi²

2019

Preprint

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A conjecture due to Cilleruelo states that for an irreducible polynomial f with integer coefficients of degree d ≥ 2, the least common multiple L f (N ) of the sequence f (1), f (2), . . . , f (N ) has asymptotic growth log L f (N ) ∼ (d − 1)N log N as N → ∞. We establish a version of this conjecture for almost all shifts of a fixed polynomial, the range of N depending on the range of shifts.

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mentioning

confidence: 79%

On Cilleruelo's conjecture for the least common multiple of polynomial sequences

Rudnick¹,

Zehavi²

2019

Preprint

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“…where we used the inequality 1 − β j ≤ αj, which follows from Remark 3.2. On the one hand, from (15) and (16)…”

Section: Lemma 41mentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

On the least common multiple of random q-integers

Sanna

2021

Res. number theory

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For every positive integer n and for every $$\alpha \in [0, 1]$$ α ∈ [ 0 , 1 ] , let $${\mathcal {B}}(n, \alpha )$$ B ( n , α ) denote the probabilistic model in which a random set $${\mathcal {A}} \subseteq \{1, \ldots , n\}$$ A ⊆ { 1 , … , n } is constructed by picking independently each element of $$\{1, \ldots , n\}$$ { 1 , … , n } with probability $$\alpha $$ α . Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $${\mathcal {A}}$$ A .Let q be an indeterminate and let $$[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]$$ [ k ] q : = 1 + q + q 2 + ⋯ + q k - 1 ∈ Z [ q ] be the q-analog of the positive integer k. We determine the expected value and the variance of $$X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )$$ X : = deg lcm ( [ A ] q ) , where $$[{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}$$ [ A ] q : = { [ k ] q : k ∈ A } . 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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“…where γ is the Euler constant and the sum is considered over all odd prime numbers. It has been proved [4] that the error term in (4) for q(x) = x 2 + 1 is O n 1/2 (log n) −4/9+ǫ for each ǫ > 0.…”

Section: The Least Common Multiple Of the Values Of A Polynomialmentioning

confidence: 99%