Almost sure convergence of least common multiple of ideals for polynomials over a number fields

For f an irreducible polynomial with integer coefficients of degree n, Cilleruelo's conjecture states that log(lcm(f(1),...,f (M))) is asymptotic to (n - 1)M log M, as M tends to infinity. 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/Q by considering the least common multiple of ideals in the ring of algebraic integers.