Let (un)n≥0 be a nondegenerate Lucas sequence satisfying un = a1un−1 +a2un−2 for all integers n ≥ 2, where a1 and a2 are some fixed relatively prime integers; and let gu be the arithmetic function defined by gu(n) := gcd(n, un), for all positive integers n. Distributional properties of gu have been studied by several authors, also in the more general context where (un)n≥0 is a linear recurrence. We prove that for each positive integer λ it holds X n ≤ x (log gu(n))λ ∼ Mu,λx as x → +∞, where Mu,λ > 0 is a constant depending only on a1, a2, and λ. More precisely, we provide an error term for the previous asymptotic formula and we show that Mu,λ can be written as an infinite series.

The moments of the logarithm of a G.C.D. related to Lucas sequences / Sanna, Carlo. - In: JOURNAL OF NUMBER THEORY. - ISSN 0022-314X. - 191:(2018), pp. 305-315. [10.1016/j.jnt.2018.03.012]

The moments of the logarithm of a G.C.D. related to Lucas sequences

Sanna, Carlo
2018

Abstract

Let (un)n≥0 be a nondegenerate Lucas sequence satisfying un = a1un−1 +a2un−2 for all integers n ≥ 2, where a1 and a2 are some fixed relatively prime integers; and let gu be the arithmetic function defined by gu(n) := gcd(n, un), for all positive integers n. Distributional properties of gu have been studied by several authors, also in the more general context where (un)n≥0 is a linear recurrence. We prove that for each positive integer λ it holds X n ≤ x (log gu(n))λ ∼ Mu,λx as x → +∞, where Mu,λ > 0 is a constant depending only on a1, a2, and λ. More precisely, we provide an error term for the previous asymptotic formula and we show that Mu,λ can be written as an infinite series.
File in questo prodotto:
File Dimensione Formato  
temp.pdf

embargo fino al 17/04/2020

Tipologia: 2. Post-print / Author's Accepted Manuscript
Licenza: Creative commons
Dimensione 305.89 kB
Formato Adobe PDF
305.89 kB Adobe PDF Visualizza/Apri
1-s2.0-S0022314X18300994-main.pdf

non disponibili

Tipologia: 2a Post-print versione editoriale / Version of Record
Licenza: Non Pubblico - Accesso privato/ristretto
Dimensione 311.13 kB
Formato Adobe PDF
311.13 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
Pubblicazioni consigliate

Caricamento pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11583/2722604