Ilaria Viglino: Almost sure convergence of least common multiple of ideals for polynomials over a number field

Abstract:

For \(f \in \mathbb{Z}[X]\) an irreducible polynomial of degree \(n\), the Cilleruelo’s conjecture states
\(\begin{equation} \log(\text{lcm}(f(1),\dots,f(M))) \sim (n – 1)M \log M \end{equation}\) as \(M \rightarrow +\infty\). The Prime Number Theorem for arithmetic progressions can be exploited to obtain an asymptotic estimate when \(f\) is a linear polynomial. Cilleruelo extended this result to quadratic polynomials. The asymptotic remains unknown for irreducible polynomials of higher degree. Recently the conjecture was shown on average for a large family of polynomials of any degree by Rudnick and Zehavi. We investigate the case of \(S_n\)-polynomials with coefficients in the ring of algebraic integers of a fixed number field extension \(K/\mathbb{Q}\) by considering the least common multiple of ideals of \(\mathcal{O}_K\).