论文标题
Prime运行功能
Prime Running Functions
论文作者
论文摘要
我们研究算术函数$φ（x; d，a）$，称为prime运行函数，其值为$ x $，总和Primes $ p_k \ equiv a \（\ text {mod} \ d）$ to $ x $和下一个prime prime $ p_ {k+1} $，最高$ x $。 （以下prime $ p_ {k+1} $可以在任何残基类$（\ text {mod} \ d）$中。）我们从经验上观察到$ x / \ log x $ in $φ（x; d，a） - φ（x; x; d，b）的系统偏见。我们为素数制定了修改的cramér模型，并表明质量间隙统计的相应总和表现出该数量级的系统偏见。将这种修饰的Cramér模型的预测与实验数据进行了比较。
We study arithmetic functions $Φ(x;d,a)$, 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 $x / \log x$ in $Φ(x;d,a) - Φ(x;d,b)$ 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.
