论文标题
Hooley的猜想的证明
A disproof of Hooley's conjecture
论文作者
论文摘要
将$ g（x; q）$定义为Primes $ p \ le x $在算术进度模型$ q $中的差异，由$ \ log p $加权。 Hooley猜想，只要$ Q $倾向于无穷大，$ x \ ge q $，我们就有上限$ g（x; q）\ ll x \ log q $。在本文中，我们表明上限一般不存在，并且$ g（x; q）$可以渐近地达到$ x（\ log q+\ log \ log \ log \ log \ log x）^2/4 $。
Define $G(x;q)$ to be the variance of primes $p\le x$ in the arithmetic progressions modulo $q$, weighted by $\log p$. Hooley conjectured that as soon as $q$ tends to infinity and $x\ge q$, we have the upper bound $G(x;q) \ll x \log q$. In this paper we show that the upper bound does not hold in general, and that $G(x;q)$ can be asymptotically as large as $x (\log q+\log\log\log x)^2/4$.
