An average version of Cilleruelo's conjecture for families of $S_n$-polynomials over a number field

Abstract

For $f\in \mathbb{Z}[X]$ an irreducible polynomial of degree $n$, the Cilleruelo's conjecture states that$$\log(\mbox{lcm}(f(1),\dots,f(M)))\sim(n-1)M\log M$$as $M\to +\infty$, where $ \mbox{lcm}(f(1),\dots,f(M)) $ is the least common multiple of $f(1),\dots ,f(M)$. 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 $S_n$-polynomials with integral coefficients over a fixed extension $K/\mathbb{Q}$ by considering the least common multiple of ideals of ${\mathcal{O}}_K$.