Shanks-Renyi素数族中的不等式：密度的一个渐近公式
Inequities in the Shanks-Renyi Prime Number Race: An asymptotic formula for the densities
摘要：
切比雪夫是第一个观察到偏见的分布素数在剩余类。一般的现象是，如果$a$是一个非平方的\mod q，而$b$是一个平方的\mod q，那么在正整数的初始区间内，与$a\mod q$全等的素数往往比与$b\mod q$全等的素数多;更简单地说，$π(x;q,a)$有超过$π(x;q,b)$的趋势。Rubinstein和Sarnak定义$δ(q;a,b)$为正实数集$x$的对数密度，这个不等式成立;直观地说，$δ(q;a,b)$是当$x$“随机选择”时，$π(x;q,a) > π(x;q,b)$的“概率”。在本文中，我们建立了一个渐近级数的$δ(q;a,b)$，可以实例化的误差项小于任何负功率的$q$。这个渐近公式是书面的方差$V(q;a,b)$，最初被定义为一个无限的总和在所有非平凡的零Dirichlet $L$ -功能对应的字符\mod q，我们如何$V(q;a,b)$可以被评估为一个有限的表达。除了提供随着$q$的增长，$δ(q;a,b)$收敛到$\frac12$的精确速率之外，这些评估还允许我们比较随着$a$和$b$以$q$为模而变化的各种密度值$δ(q;a,b)$;通过分析得到的公式，我们可以基于剩余类$a$和$b\mod q$的算术性质来解释和预测这些密度中的哪一个将更大或更小。例如，我们表明，如果$a$是一个素数的权力和$a'$不是，然后$δ(q;a,1) < δ(q;a',1)$的所有，但许多模$q$的$a$和$a'$都是非平方。最后，我们建立了严格的数值界限，这些密度$δ(q;a,b)$和报告的广泛计算。
英文摘要：
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 $b\mod q$ in initial intervals of the positive integers; more succinctly, there is a tendency for $π(x;q,a)$ to exceed $π(x;q,b)$. Rubinstein and Sarnak defined $δ(q;a,b)$ to be the logarithmic density of the set of positive real numbers $x$ for which this inequality holds; intuitively, $δ(q;a,b)$ is the "probability" that $π(x;q,a) > π(x;q,b)$ when $x$ is "chosen randomly". In this paper, we establish an asymptotic series for $δ(q;a,b)$ that can be instantiated with an error term smaller than any negative power of $q$. This asymptotic formula is written in terms of a variance $V(q;a,b)$ that is originally defined as an infinite sum over all nontrivial zeros of Dirichlet $L$-functions corresponding to characters\mod q; we show how $V(q;a,b)$ can be evaluated exactly as a finite expression. In addition to providing the exact rate at which $δ(q;a,b)$ converges to $\frac12$ as $q$ grows, these evaluations allow us to compare the various density values $δ(q;a,b)$ as $a$ and $b$ vary modulo $q$; by analyzing the resulting formulas, we can explain and predict which of these densities will be larger or smaller, based on arithmetic properties of the residue classes $a$ and $b\mod q$. For example, we show that if $a$ is a prime power and $a'$ is not, then $δ(q;a,1) < δ(q;a',1)$ for all but finitely many moduli $q$ for which both $a$ and $a'$ are nonsquares. Finally, we establish rigorous numerical bounds for these densities $δ(q;a,b)$ and report on extensive calculations of them.
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