Related papers: On the density of primes in arithmetic progression…

For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set…

Let $p>2$ be prime and $g$ a primitive root modulo $p$. We present an argument for the fact that discrete logarithms of the numbers in any arithmetic progression are uniformly distributed in $[1,p]$ and raise some questions on the subject.

For a primitive Hilbert modular form $f$ over $F$ of weight $k$, under certain assumptions on image of $\bar{\rho}_{f,\lambda}$, we calculate the Dirichlet density of primes $\mathfrak{p}$ for which the $\mathfrak{p}$-th Fourier coefficient…

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…

Let $x\geq 1$ be a large number, and let $1 \leq a <q $ be integers such that $\gcd(a,q)=1$ and $q=O(\log^c)$ with $c>0$ constant. This note proves that the counting function for the number of primes $p \in \{p=qn+a: n \geq1 \}$ with a…

For a given finitely generated multiplicative subgroup of the rationals which possibly contain negative numbers, we derive, subject to GRH, formulas for the densities of primes for which the index of the reduction group has a given value.…

Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$ such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic density relative to the set of all primes which is at least $\prod_{i=1}^k…

Let N_g(d) be the set of primes p such that the order of g modulo p is divisible by a prescribed integer d. Wiertelak showed that this set has a natural density and gave a rather involved explicit expression for it. Let N_g(d)(x) be the…

We extend the character sum method for the computation of densities in Artin primitive root problems developed by H. W. Lenstra and the authors to the situation of radical extensions of arbitrary rank. Our algebraic set-up identifies the…

Let K be a number field, and let a be a non-zero element of K. Fix some prime number l. We compute the density of the following set: the primes p of K such that the multiplicative order of the reduction of a modulo p is coprime to l (or,…

We present in this work a heuristic expression for the density of prime numbers. Our expression leads to results which possesses approximately the same precision of the Riemann's function in the domain that goes from 2 to 1010 at least.…

We prove that if $A$ is any set of prime numbers satisfying \[ \sum_{a\in A}\frac{1}{a}=\infty, \] then $A$ must contain a $3$-term arithmetic progression. This is accomplished by combining the transference principle with a density…

A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p-1$ whose inverse $\bar{a}$ within the same range has opposite parity. Lehmer numbers that are also primitive roots have been discussed by Wang and Wang in an…

Let \(u\neq \pm 1,v^2\) be a fixed integer, let \(p\geq 2\) be a prime, and let $\text{ord}_p(u) \mid p-1$ be the multiplicative order of $u \text{ mod } p$. Define a prime counting function by $\pi(u,x)=\# \{ p\leq x:\text{ord}_p(u)=p-1…

We generalize a theorem of Nymann that the density of points in Z^d that are visible from the origin is 1/zeta(d), where zeta(a) is the Riemann zeta function 1/1^a + 1/2^a + 1/3^a + ... A subset S of Z^d is called primitive if it is a…

In this paper we show that if $A$ is a subset of the primes with positive relative density $\delta$, then $A+A$ must have positive upper density $C_1\delta e^{-C_2(\log(1/\delta))^{2/3}(\log\log(1/\delta))^{1/3}}$ in $\mathbb{N}$. Our…

Let g be a non-zero rational number. Let N_{g,t}(x) denote the number of primes p<=x for which the subgroup of the multiplicative group of the finite field having p elements that is generated by g mod p is of residual index t. In Part I,…

Let $A\subset\left\{ 1,\dots,N\right\} $ be a set of prime numbers containing no non-trivial arithmetic progressions. Suppose that $A$ has relative density $\alpha=|A|/\pi(N)$, where $\pi(N)$ denotes the number of primes in the set $\left\{…

Given a pair of distinct non-CM normalized eigenforms having integer Fourier coefficients $a_1 (n)$ and $a_2(n)$, we count positive integers $n$ with $(a_1(n), a_2(n))=1$ and make a conjecture about the density of the set of primes $p$ for…

In 1927, E. Artin proposed a conjecture for the natural density of primes $p$ for which $g$ generates $(\mathbb{Z}/p\mathbb{Z})^\times$. By carefully observing numerical deviations from Artin's originally predicted asymptotic, Derrick and…