论文标题
在残留类中的素数之间的差距的第一次发生
On the first occurrences of gaps between primes in a residue class
论文作者
论文摘要
我们研究了算术进程（p）中素数之间差距的第一次出现：$ r $，$ r+q $，$ r+2q $，$ r+3q，\ ldots，$ q $ $ q $和$ r $是coprime Integers，$ q> r \ ge1 $。第一出现差距大小的生长趋势和分布与（p）中素数之间的最大差距相似。在适当重新进行恢复后，第一出现差距大小的直方图与Gumbel极值分布相吻合。计算表明，第一出现差距比最大差距要多得多：$ o（\ log^2 x）$在$ x $以下（p）中的数量之间存在$ o（\ log^2 x）$，而最大差距的数量仅为$ o（\ log x）$。我们探索给定尺寸间隙的渐近密度与布伦常数的相应概括之间的联系。对于第一次出现（p）的gap $ d $，我们期望空白终止元素$ p \ asymp \ sqrt {d}} \ exp（\ sqrt {\ sqrt {d/φ（q）}）$ cobs $ complyise频繁经常。最后，我们在第一次出现间隙序列中研究了间隙大小作为其指数的函数。
We study the first occurrences of gaps between primes in the arithmetic progression (P): $r$, $r+q$, $r+2q$, $r+3q,\ldots,$ where $q$ and $r$ are coprime integers, $q>r\ge1$. The growth trend and distribution of the first-occurrence gap sizes are similar to those of maximal gaps between primes in (P). The histograms of first-occurrence gap sizes, after appropriate rescaling, are well approximated by the Gumbel extreme value distribution. Computations suggest that first-occurrence gaps are much more numerous than maximal gaps: there are $O(\log^2 x)$ first-occurrence gaps between primes in (P) below $x$, while the number of maximal gaps is only $O(\log x)$. We explore the connection between the asymptotic density of gaps of a given size and the corresponding generalization of Brun's constant. For the first occurrence of gap $d$ in (P), we expect the end-of-gap prime $p\asymp\sqrt{d}\exp(\sqrt{d/φ(q)})$ infinitely often. Finally, we study the gap size as a function of its index in the sequence of first-occurrence gaps.
