Related papers: On Artin's Primitive Root Conjecture for Function …

In 1927, E. Artin conjectured that all non-square integers $a\neq -1$ are a primitive root of $\mathbb{F}_p$ for infinitely many primes $p$. In 1967, Hooley showed that this conjecture follows from the Generalized Riemann Hypothesis (GRH).…

We consider Artin's conjecture on primitive roots over a number field $K$, reducing an algebraic number $\alpha\in K^\times$. Under the Generalised Riemann Hypothesis, there is a density ${\mathrm{dens}}(\alpha)$ counting the proportion of…

An integer is a primitive root modulo a prime $p$ if it generates the whole multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$. In 1927 Artin conjectured that an integer $a$ which is not $-1$ or a square is a primitive root for infintely…

Let a and b be non-zero rational numbers that are multiplicatively independent. We study the natural density of the set of primes p for which the subgroup of the multiplicative group of the finite field with p elements generated by (a\mod…

The celebrated Artin conjecture on primitive roots asserts that given any integer $g$ which is neither $-1$ nor a perfect square, there is an explicit constant $A(g)>0$ such that the number $\Pi(x;g)$ of primes $p\le x$ for which $g$ is a…

A famous conjecture of Artin asserts that any integer $a$ that is neither $-1$ nor a square should be a primitive root (mod $p$) for a positive proportion of primes $p$. Moreover, using a heuristic argument, Artin guessed an explicit…

It follows from the work of Artin and Hooley that, under assumption of the generalized Riemann hypothesis, the density of the set of primes $q$ for which a given non-zero rational number $r$ is a primitive root modulo $q$ can be written as…

Fix an integer $g \neq -1$ that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which $g$ is a primitive root. Forty years later, Hooley showed that Artin's conjecture follows from the…

This work concerns Artin's Conjecture on primitive roots and related problems for number fields. Let $K$ be a number field and let $W_1$ to $W_n$ be finitely generated subgroups of $K^\times$ of positive rank. We consider the index map,…

Using techniques of algebraic and analytic number theory, we resolve a question on monoid rings posed by Kulosman, et. al., under the assumption of the Generalized Riemann Hypothesis (GRH). Specifically, we show that under an appropriate…

Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\times$. For all but finitely many primes $\mathfrak p$ of $K$, the reduction $(G \bmod \mathfrak p)$ generates a well-defined subgroup of the multiplicative group…

E. Artin conjectured that any integer $a >1$ which is not a perfect square is a primitive root modulo $p$ for infinitely many primes $p.$ Let $f_a(p)$ be the multiplicative order of the non-square integer $a$ modulo the prime $p.$ M. R.…

Let a and f be coprime positive integers. Let g be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes p such that p=a(mod f) and g is a primitive root modulo p has a…

We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p+1. An extension of…

Let a,f and g be integers, with a and f coprime. Under the generalized Riemann hypothesis it follows from work of Hooley and Lenstra that the set of primes p such that p=a(mod f) and g is primitive root mod p has a natural density. In this…

Let $\Gamma\subset\mathbb{Q}^*$ be a finitely generated subgroup and let $p$ be a prime such that the reduction group $\Gamma_p$ is a well defined subgroup of the multiplicative group $\mathbb{F}_p^*$. We prove an asymptotic formula for the…

For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In…

Erd\H{o}s proved that $\mathcal{F}(A) := \sum_{a \in A}\frac{1}{a\log a}$ converges for any primitive set of integers $A$ and later conjectured this sum is maximized when $A$ is the set of primes. Banks and Martin further conjectured that…

Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil…

Since Hooley's seminal 1967 resolution of Artin's primitive root conjecture under the Generalized Riemann Hypothesis, numerous variations of the conjecture have been considered. We present a framework generalizing and unifying many…