Hardy-Ramanujan Journal |

E. Artin conjectured that any integer $a > 1$ which is not a perfect square is a primitive root modulo $p$ for infinitely many primes $ p.$ Let $f_a(p)$ be the multiplicative order of the non-square integer $a$ modulo the prime $p.$ M. R. Murty and S. Srinivasan \cite{Murty-Srinivasan} showed that if $\displaystyle \sum_{p < x} \frac 1 {f_a(p)} = O(x^{1/4})$ then Artin's conjecture is true for $a.$ We relate the Murty-Srinivasan condition to sums involving the cyclotomic periods from the subfields of $\mathbb Q(e^{2\pi i /p})$ corresponding to the subgroups $<a> \subseteq \mathbb F_p^*.$

Source : oai:HAL:hal-03251183v2

Volume: Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021

Published on: January 9, 2022

Accepted on: January 9, 2022

Submitted on: July 10, 2021

Keywords: primitive roots,cyclotomic periods,exponential sums,11A07, 11R18, 11T23, 11L07,[MATH]Mathematics [math]

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