The least common multiple of a bivariate quadratic sequence

DOI10.4064/AA220719-9-7arXiv2206.05817OpenAlexW4388026741MaRDI QIDQ6071871

Author name not available (Why is that?)

Publication date: 29 November 2023

Published in: Acta Arithmetica (Search for Journal in Brave)

Abstract: Let

F i n m a t h b b Z [x, y]

be some polynomial of degree 2. In this paper we find the asymptotic behaviour of the least common multiple of the values of

F

up to

N

. More precisely, we consider

p s i_{F} (N) = l o g l e f t (e x t {L C M}_{0 < F (x, y) l e q N} l e f t l b r a c e F (x, y) i g h t b r a c e i g h t)

as

N

tends to infinity. It turns out that there are 4 different possible asymptotic behaviours depending on

F

. For a generic

F

, we show that the function

p s i_{F} (N)

has order of magnitude

f r a c N l o g l o g N s q r t l o g N

. We also show that this is the expected order of magnitude according to a suitable random model. However, special polynomials

F

can have different behaviours, which sometimes deviate from the random model. We give a complete description of the order of magnitude of these possible behaviours, and when each one occurs.

Full work available at URL: https://arxiv.org/abs/2206.05817

zbMATH Keywords

quadratic forms \(\mathrm{LCM}\)half-dimensional sieve

Mathematics Subject Classification ID

Applications of sieve methods (11N36) Primes represented by polynomials; other multiplicative structures of polynomial values (11N32)