**Speaker**

*Ilaria Viglino, EPFL*

**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 \( \mathcal{O}_K \).