Related papers: Chebyshev's bias without linear independence

In this paper we produce unconditionally new instances of Galois number field extensions exhibiting strong discrepancies in the distribution of Frobenius elements among conjugacy classes of the Galois group. We first prove an inverse Galois…

Let $\pi(x;q,a)$ denote the number of primes up to $x$ that are congruent to $a$ modulo $q$. A prime number race, for fixed modulus $q$ and residue classes $a_1, \ldots, a_r$, investigates the system of inequalities $\pi(x;q,a_1) >…

We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. Our best results is conditional on the Generalised Riemann hypothesis (GRH). As a…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication. For a prime $p$ of good reduction for $E$, we write $\#E_p(\mathbb{F}_p) = p + 1 - a_p(E)$ for the number of $\mathbb{F}_p$-rational points of the…

We establish an unconditional effective Chebotarev density theorem that improves uniformly over the well-known result of Lagarias and Odlyzko. As a consequence, we give a new asymptotic form of the Chebotarev density theorem that can count…

Rubinstein and Sarnak have shown, conditional on the Riemann hypothesis (RH) and the linear independence hypothesis (LI) on the non-real zeros of $\zeta(s)$, that the set of real numbers $x\ge2$ for which $\pi(x)>$ li$(x)$ has a logarithmic…

Fix a prime $p >2$ and a finite field $\mathbb{F}_{q}$ with $q$ elements, where $q$ is a power of $p$. Let $m$ be a monic polynomial in the polynomial ring $\mathbb{F}_{q}[T]$ such that $deg(m)$ is large. Fix an integer $r\geq 2$, and let…

Rubinstein and Sarnak investigated systems of inequalities of the form pi(x;q,a_1) > ... > pi(x;q,a_r), where pi(x;q,a) denotes the number of primes up to x that are congruent to a mod q. They showed, under standard hypotheses on the zeros…

We improve the Chebotarev variant of the Brun-Titchmarsh theorem proven by Lagarias, Montgomery, and Odlyzko using the log-free zero density estimate and zero repulsion phenomenon for Hecke L-functions that were recently proved by the…

We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form h(n) for natural numbers n, or h(p) with p prime, for appropriate polynomials h with integer coefficients. The…

We establish an explicit bound for the least prime occurring in the Chebotarev density theorem without any restriction. Let $L/K$ be any Galois extension of number fields such that $L\not=\mathbb{Q}$, and let $C$ be a conjugacy class in the…

For a fixed number field $K$, we consider the mean square error in estimating the number of primes with norm congruent to $a$ modulo $q$ by the Chebotar\"ev Density Theorem when averaging over all $q\le Q$ and all appropriate $a$. Using a…

In this article, we prove that an asymptotic formula for the prime number race with respect to Fermat curves of prime degree is equivalent to part of the Deep Riemann Hypothesis (DRH), which is a conjecture on the convergence of partial…

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois…

Fix an integer $r\geq 3$. Let $q$ be a large positive integer and $a_1,...,a_r$ be distinct residue classes modulo $q$ that are relatively prime to $q$. In this paper, we establish an asymptotic formula for the logarithmic density…

In this paper we investigate higher moments attached to the Chebotarev Density Theorem. Our focus is on the impact that peculiar Galois group structures have on the limiting distribution. Precisely we consider in this paper the case of…

We study the asymptotic estimation of prime ideals that satisfy certain congruence and argument conditions in imaginary quadratic fields. We also discuss the phenomenon of Chebyshev's bias in the distribution of prime ideals among different…

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb Z/p\mathbb Z)^*,$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each primitive residue class contains a…

We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call "complete bias",…

We provide new sufficient conditions for Chebyshev estimates for Beurling generalized primes. It is shown that if the counting function $N$ of a generalized number system satisfies the $L^{1}$-condition $$…