Abstract

We study arithmetic functions $\Phi (x;d,a)$, called prime running functions, whose value at x sums the gaps between primes $p_k\equiv a(\text{mod}d)$ below x and the next following prime $p_{k+1}$, up to x. (The following prime $p_{k+1}$ may be in any residue class $(\text{mod}d)$.) We empirically observe systematic biases of order $x/\hspace{0.17em}\text{log}\hspace{0.17em}x$ in $\Phi (x;d,a)-\Phi (x;d,b)$ for different a, b. We formulate modified Cramér models for primes and show that the corresponding sum of prime gap statistics exhibits systematic biases of this order of magnitude. The predictions of such modified Cramér models are compared with the experimental data.

中文翻译：

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主要运行函数

摘要

我们研究算术函数$\Phi (X;d,\mathrm{一个})$，称为素数运行函数，其在x处的值求和素数之间的差距$p_k\equiv \mathrm{一个}(\text{模组}d)$低于x和下一个素数$p_{k+\mathrm{1个}}$, 直到x。（下面素数$p_{k+\mathrm{1个}}$可能属于任何残基类别$(\text{模组}d)$.) 我们凭经验观察到系统的秩序偏差$X/\hspace{0.17em}\text{日志}\hspace{0.17em}X$在$\Phi (X;d,\mathrm{一个})-\Phi (X;d,b)$对于不同的 a，b。我们为素数制定了修正的 Cramér 模型，并表明相应的素数间隙统计总和表现出这个数量级的系统偏差。将此类修改后的 Cramér 模型的预测与实验数据进行了比较。