
TL;DR
This paper investigates bounds on primitive roots modulo primes, improving estimates for the least primitive roots and prime primitive roots, and provides lower bounds on primes with a fixed primitive root, advancing understanding in number theory.
Contribution
It offers sharper bounds for the least primitive roots and prime primitive roots, and establishes a new lower bound on the count of primes with a fixed primitive root, improving previous results.
Findings
Sharpened estimate g(p) <= p^(5/loglog p) for large primes
Lower bound #{p <= x : ord(g)= p-1} >> x/log x for fixed primitive roots
Improved understanding of primitive roots distribution and bounds
Abstract
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) <<(log p)^c, c>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, b^2 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…
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Taxonomy
TopicsAnalytic Number Theory Research · Coding theory and cryptography · Mathematics and Applications
