2022/12/12 by Carlo Sanna, Sanna, Carlo
Mathematics · Physics and Astronomy · #11B39 (Primary) 11N05 #11N37 (Secondary) #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2212.06127
openalex publication_date 2022/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let u = (un)n ≥ 0 be a Lucas sequence, that is, a sequence of integers satisfying u0 = 0, u1 = 1, and un = a1 un - 1 + a2 un - 2 for every integer n ≥ 2, where a1 and a2 are fixed nonzero integers. For each prime number p with p \nmid 2a2Du, where Du := a12 + 4a2, let ρu(p) be the rank of appearance of p in u, that is, the smallest positive integer k such that p | uk. It is well known that ρu(p) exists and that p ≡ (Du | p ) \pmod ρu(p), where (Du | p ) is the Legendre symbol. Define the index of appearance of p in u as ιu(p) := (p - (Du | p )) / ρu(p). For each positive integer t and for every x > 0, let Pu(t, x) be the set of prime numbers p such that p ≤ x, p \nmid 2a2 Du, and ιu(p) = t. Under the Generalized Riemann Hypothesis, and under some mild assumptions on u, we prove that #Pu(t, x) = A Fu(t) Gu(t) (x)/(log x) + Ou ((x)/((log x)2) + (x log (2log x))/(φ(t) (log x)2)) , for all positive integers t and for all x > t3, where A is the Artin constant, Fu(⋅) is a multiplicative function, and Gu(⋅) is a periodic function (both these functions are effectively computable in terms of u). Furthermore, we provide some explicit examples and numerical data.