On the Least Common Multiple of Polynomials over a Number Field

For an irreducible integral polynomial f of degree n, Cilleruelo’s conjecture states an asymptotic formula for the logarithm of the least common multiple of the first M values f(1) to f(M). It’s well-known for n = 1 as a consequence of Dirichlet’s Theorem for primes in arithmetic progression, and it was proved by Cilleruelo for quadratic polynomials. Recently the conjecture was shown by Rudnick and Zehavi for a large family of polynomials of any degree. We want to investigate an average version of the conjecture for Sn-polynomials with integral coefficients over a fixed extension K=Q by considering the least common multiple of ideals of OK. The case of linear polynomials is dealt with separately by exploiting Dirichlet’s Theorem for primes in arithmetic progression, to get an asymptotic estimate. In our case, to achieve explicit error terms, we want effective versions of the asymptotics. We will state here both a conditional and unconditional results proved by Lagarias and Odlyzko. For degree-2 polynomials, it is possible to obtain explicit asymptotics for the least common multiple, analogously to the ones achieved for polynomials in Z[X]. However, the latter is not a subject of the current paper.