Biases in the Shanks—Rényi Prime Number Race
Biases in the Shanks—Rényi Prime Number Race
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Shanks 中的偏差——Rényi 素数竞赛
作者：
Andrey Feuerverger;G. Martin
鲁宾斯坦和萨纳克研究了形式为π(x；q，a1)>；…的不等式组>；π(x；q，Ar)，其中p(x；q，b)表示与bmod q同余的素数，p(x；q，b)在关于狄里克莱特L函数mod q的零点的标准假设下，证明了这些不等式成立的一组正实数x具有正的(对数)密度δq；al，….AR&>0。他们还发现了一个令人惊讶的事实，即与这些密度相关的某个分布在剩余类Aj的排列下一般不是对称的，即使Aj都是平方或都是非平方mod Q(这是避免切比雪夫首先观察到的那种类型的明显偏差的必要条件)。这种不对称性表明，与先前的预期相反，密度δq；al，…，AR本身在AJ的排列下有所不同。在这里，我们(在鲁宾斯坦和萨纳克使用的假设下)推导出密度δq；al，…的一般公式对于特殊的模q=8和q=12，以及对非平方的排列{3，5，7}mod 8和{5，7，11}mod 12，分别对{al，a2，a3}，我们严格地限制了计算中的误差，从而验证了这些密度在Aj的排列下确实是不对称的。我们还确定了密度δq；al，…的几种情况在AJ的某些排列下，AR保持不变，在某些情况下，它们可以被证明是不同的。
Rubinstein and Sarnak investigated systemsof inequalities of the form π(x; q, a1) > … > π(x; q, ar), where p(x; q, b) denotes the number of primes up to x that are congruent to b mod q. They showed, under standard hypotheses on the zeros of Dirichlet L-functions mod q, that the set of positive real numbers x for which these inequalities hold has positive (logarithmic) density δq;al, … .ar > 0. They also discovered the surprising fact that a certain distribution associated with these densities is not symmetric under permutations of the residue classes aj in general, even if the aj are all squaresor all nonsquares mod q (a condition necessary to avoid obvious biases of the type first observed by Chebyshev). This asymmetry suggests, contrary to prior expectations, that the densities δq;al , …,ar themselves vary under permutations of the aj. Here we derive (under the hypotheses used by Rubinstein and Sarnak) a general formula for the densities δq;al , …,ar, and We use this formula to calculate many of these densities when q ≤ 12 and r ≤ 4. For the special moduli q = 8 and q = 12, and for {al, a2,a3} a permutation of the nonsquares {3, 5, 7} mod 8 and {5, 7, 11} mod 12, respectively, we rigorously bound the error in our calculations, thus verifying that these densities are indeed asymmetric under permutation of the aj. We also determine several situations in which the densities δq;al , …, ar remain unchanged under certain permutations of the aj, and some situations in which they are provably different.