2014/05/01 by N. A. Carella, Carella, N. A.
Computer Science · Mathematics · #11M26 #11N37 #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #General Mathematics (math.GM) #Mathematics and Applications #Primary 11A07 #Secondary 11Y16
paper · pdf · doi:10.48550/arxiv.1405.0161
openalex publication_date 2014/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This monograph considers a few topics in the theory of primitive roots g(p) modulo a prime p>=2. A few estimates of the least primitive roots g(p) and the least prime primitive roots g^*(p) modulo p, a large prime, are determined. One of the estimate here seems to sharpen the Burgess estimate g(p) << p^(1/4+e) for arbitrarily small number 3 > 0, to the smaller estimate g(p) <= p^(5/loglog p) uniformly for all large primes p => 2. The expected order of magnitude is g(p) <1 constant. The corresponding estimates for least prime primitive roots g^*(p) are slightly higher. Anotrher topic deals with an effective lower bound #p <= x : ord(g)= p-1 >> x/log x for the number of primes p <= x with a fixed primitive root g != -1, b2 for all large number x >1. The current results in the literature claim the lower bound #p <= x : ord(g) = p-1 >> x/(log x)2, and have restrictions on the minimal number of fixed integers to three or more.