Simultaneous Primitive Roots over Finite Rings

TL;DR

The paper investigates the average density of primes $p$ in $[x,2x]$ for which a primitive root modulo $p$ lifts to a primitive root modulo $p^2$, i.e., stationary primitive roots, and contrasts them with nonstationary cases. It builds a framework of simultaneous characteristic functions to count stationary versus nonstationary primes, derives a leading term $N_s(x,z)\sim c_2 z^2 x/\log x$ and a complementary $N_n(x,z)\sim c_3 z^2 x/\log x$ with $z= p^{1/2+\varepsilon}\log p$, and provides explicit error bounds via exponential-sum estimates. The constants $c_2$ and $c_3$ are expressed through Artin-type constants, linking to totient averages over shifted primes, while sections on least stationary primitive roots and random-stationary densities establish broader structural results and positivity of the average density. The work culminates in a set of open problems and conjectures, including rational expansions and prime-divisor sums, indicating rich connections to Artin-type phenomena, Wieferich-like behavior, and shifted-prime arithmetic.

Abstract

This note investigates the average density of prime numbers $p\in[x,2x]$ with respect to a random simultaneous primitive root $g\leq p^{1/2+\varepsilon}$ over the finite rings $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}/p^2\mathbb{Z}$ as $x \to \infty$.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.1
• Definition 2.1
• Definition 2.2
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• proof : Proof