We generalize current known distribution results on Shanks--Renyi 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 field analogues developed in the recent years. More precisely, let $\pi(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo $q$. For a fixed integer $q$ and distinct invertible congruence classes $a_0,a_1,\ldots,a_D$, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of $x$ real for which the inequalities $\pi(x;q,a_0)>\pi(x;q,a_1)> \ldots >\pi(x;q,a_D)$ are simultaneously satisfied admits a logarithmic density.

中文翻译：

---

限制任意大量残基类中素数分布的特性

我们将当前已知的 Shanks-Renyi 素数竞赛分布结果推广到涉及任意多个残基类别的情况。我们的方法处理了可以追溯到切比雪夫的经典案例和近年来开发的函数场类似物。更准确地说，令 $\pi(x;q,a)$ 是直到 $x$ 与 $a$ 模 $q$ 全等的素数的数目。对于固定整数 $q$ 和不同的可逆同余类 $a_0,a_1,\ldots,a_D$，假设广义黎曼假设和线性独立假设的弱版本，我们证明不等式 $\pi(x;q,a_0)>\pi(x;q,a_1)> \ldots >\pi(x;q,a_D)$ 同时满足承认对数密度。