Densities Of Primes And Primitive Roots

zbMATH Open1395.11008arXiv1707.06517MaRDI QIDQ2960916

Nelson A. Carella

Publication date: 17 February 2017

Abstract: Let $ueqpm1,v^2$ be a fixed integer, let $pgeq2$ be a prime, and let $ext{ord}_p(u)=d:|:p-1$ be the order of $uextmodp$. This note provides an effective lower bound $$ for the number of primes $pleqx$ with a fixed primitive root $uextmodp$ for all large numbers $xgeq1$. The current results in the literature have the lower bound $$, and restrictions on the fixed primitive root to a subset of at least three or more integers. An application to repeating decimal $1/p$ of maximal period $p-1$ is included, and a precise counting function for the number of primes $pleqx$ with a fixed primitive root $10extmodp$ for all large numbers $xgeq1$.

Full work available at URL: https://arxiv.org/abs/1707.06517

zbMATH Keywords

primitive rootprime numberArtin primitive root

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37) Congruences; primitive roots; residue systems (11A07)

Related Items (5)

Densities of primes and realization of local extensions ⋮ Densities of Sets of Primes Related to Decimal Expansion of Rational Numbers ⋮ On gigantic density of zeros of some signals defined by prime numbers ⋮ Character sums for primitive root densities ⋮ Densities in certain three-way prime number races

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