相关论文: Biases in the Shanks-Renyi Prime Number Race

Chebyshev was the first to observe a bias in the distribution of primes in residue classes. The general phenomenon is that if $a$ is a nonsquare\mod q and $b$ is a square\mod q, then there tend to be more primes congruent to $a\mod q$ than…

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…

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…

It has been well-observed that an inequality of the type $\pi(x;q,a) > \pi(x;q,b)$ is more likely to hold if $a$ is a non-square modulo $q$ and $b$ is a square modulo $q$ (the so-called ``Chebyshev Bias''). For instance, each of…

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 generalize current known distribution results on Shanks--R\'enyi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function…

Under certain plausible assumptions, M. Rubinstein and P. Sarnak solved the Shanks--R\'enyi race problem, by showing that the set of real numbers $x\geq 2$ such that $\pi(x;q,a_1)>\pi(x;q,a_2)>...>\pi(x;q,a_r)$ has a positive logarithmic…

Let $q\geq 3$, $2\leq r\leq \phi(q)$ and $a_1,...,a_r$ be distinct residue classes modulo $q$ that are relatively prime to $q$. Assuming the Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis, M. Rubinstein and P. Sarnak…

Let a be a positive integer greater than 1, and Q_a(x;k,j) be the set of primes p less than x such that the residual order of a(mod p) is congruent to j modulo k. In this paper, the natural densities of Q_a(x;4,j) (j=0,1,2,3) are…

Let $a_1$, $a_2$, and $a_3$ be distinct reduced residues modulo $q$ satisfying the congruences $a_1^2 \equiv a_2^2 \equiv a_3^2 \pmod q$. We conditionally derive an asymptotic formula, with an error term that has a power savings in $q$, for…

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…

Chebyshev observed in a letter to Fuss that there tends to be more primes of the form $4n+3$ than of the form $4n+1$. The general phenomenon, which is referred to as Chebyshev's bias, is that primes tend to be biased in their distribution…

Let a be a positive integer which is not a perfect h-th power with h greater than 1, and Q_a(x;4,j) be the set of primes p less than x such that the residual order of a(mod p) is congruent to j modulo 4. When j=0, 2, it is known that…

For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $\varphi(q)$ reduced…

Under two assumptions, we determine the distribution of the difference between two functions each counting the numbers < x that are in a given arithmetic progression modulo q and the product of two primes. The two assumptions are (i) the…

We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus $q$, any two reduced congruence classes $a_1$ and $a_2$ mod $q$, and any $r_1,r_2 \ge 1$, a positive…

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…

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 $\pi_{q,a}(x)$ denote the number of primes $\le x$ in the progression $a$ modulo $q$. We study subtle inequities in these functions, with $q$ fixed and variable $a$ (sometimes called 'prime race problems'). It is known unconditionally…

Chebyshev famously observed empirically that more often than not, there are more primes of the form $3 \bmod 4$ up to $x$ than of the form $1 \bmod 4$. This was confirmed theoretically much later by Rubinstein and Sarnak in a logarithmic…