Title:  Artin’s primitive root conjecture and the index of the reductions of algebraic numbers

Speaker:   Pietro Sgobba  (Associate Professor)

Time: Tuesday, January 18, 2024, AM: 10:00-11:00

Venue: Room 3A01, Building No. 37, Wushan Campus

Abstract: Artin’s primitive root conjecture (1927) states that given an integer a which is neither 0,1,-1, nor a perfect square, there are infinitely many primes p such that a is a primitive root modulo p. We consider a more general question over number fields. Let K be a number field and let a be a nonzero algebraic number in K. For all but finitely many primes p of K, the reduction (a mod p) is well-defined, and we may consider its multiplicative index. Assuming the Generalized Riemann Hypothesis, we then study the natural density of primes of K for which this index lies in a given set of positive integers, and we focus on the case where this set is defined by prescribing valuations for its elements. The obtained results hold unconditionally under certain conditions. Part of the work is joint with Järviniemi, Perucca and Moree.