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Densities Of Primes And Primitive Roots

N. A. Carella

TL;DR

The work analyzes primes $p$ for which a fixed integer $u$ is a primitive root modulo $p$, establishing an effective lower bound $ obreak \\\pi_u(x) \\gg \\frac{x}{\\log x}$ and clarifying the density $\\delta(u)$ of such primes. It develops divisors-free representations for the primitive-root indicator and leverages powerful exponential-sum bounds, including finite summation kernels and Gauss sums, to control error terms. The main term is linked to the Artin constant $a_1$ via sums of $\\varphi(p-1)$ and related averages, yielding unconditional positive densities and precise asymptotics for bases such as $u=10$, where $\\delta(10)=a_1 \\\approx 0.3739...$, and $\\pi_{10}(x)=\\delta(10) \\mathrm{li}(x) + O(x/\\log^B x)$. The results generalize to maximal-period decimal expansions and fixed-base counts, providing stronger density results than prior $x/\\log^2 x$ bounds and removing subset restrictions on $u$, with implications for Artin-type conjectures on primes and primitive roots. The methodology offers a versatile framework for estimating primitive-root densities via divisors-free indicators and robust exponential-sum techniques, enhancing our understanding of primitive-root distributions in the integers modulo primes.

Abstract

Let $u\neq \pm 1,v^2$ be a fixed integer, let $p\geq 2$ be a prime, and let $\text{ord}_p(u)=d \:|\: p-1$ be the order of $u \text{ mod } p$. This note provides an effective lower bound $π_u(x)=\# \{ p\leq x:\text{ord}_p(u)=p-1 \}\gg x (\log x)^{-1}$ for the number of primes $p\leq x$ with a fixed primitive root $u \text{ mod } p$ for all large numbers $x\geq 1$. The current results in the literature have the lower bound $π_u(x)=\# \{ p\leq x:\text{ord}_p(u)=p-1 \}\gg x (\log x)^{-2}$, 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 $p\leq x$ with a fixed primitive root $10 \text{ mod } p$ for all large numbers $x\geq 1$.

Densities Of Primes And Primitive Roots

TL;DR

The work analyzes primes for which a fixed integer is a primitive root modulo , establishing an effective lower bound and clarifying the density of such primes. It develops divisors-free representations for the primitive-root indicator and leverages powerful exponential-sum bounds, including finite summation kernels and Gauss sums, to control error terms. The main term is linked to the Artin constant via sums of and related averages, yielding unconditional positive densities and precise asymptotics for bases such as , where , and . The results generalize to maximal-period decimal expansions and fixed-base counts, providing stronger density results than prior bounds and removing subset restrictions on , with implications for Artin-type conjectures on primes and primitive roots. The methodology offers a versatile framework for estimating primitive-root densities via divisors-free indicators and robust exponential-sum techniques, enhancing our understanding of primitive-root distributions in the integers modulo primes.

Abstract

Let be a fixed integer, let be a prime, and let be the order of . This note provides an effective lower bound for the number of primes with a fixed primitive root for all large numbers . 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 of maximal period is included, and a precise counting function for the number of primes with a fixed primitive root for all large numbers .

Paper Structure

This paper contains 21 sections, 18 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Exponentials and Character Sums
  3. Simple Characters Sums
  4. Representations of the Characteristic Functions
  5. Primitive Roots Tests
  6. Divisors Dependent Characteristic Function
  7. Divisors Free Characteristic Function
  8. Finite Summation Kernel and Gauss Sum
  9. Finite Summation Kernel
  10. Gauss Sum
  11. Estimates of Power Exponential Sums
  12. Power Exponential Sum with Relatively Prime Index
  13. FFT of Power Exponential Sum with Relatively Prime Index
  14. FFT of Power Exponential Sum with Relatively Prime Index
  15. Evaluation of the Main Term
  16. ...and 6 more sections

Key Result

Theorem 1.1

A fixed integer $u\neq \pm 1,v^2$ is a primitive root mod $p$ for infinitely many primes $p\geq 2$. In addition, the density of these primes satisfies where $\mathop{\mathrm{li}}\nolimits(x)$ is the logarithm integral, and $\delta(u)>0$ is a constant, for all large numbers $x\geq 1$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Definition 3.1
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.2
  • Lemma 3.3
  • proof : Proof
  • Lemma 4.1
  • ...and 22 more