论文标题
关于贝特曼和钻石的猜想的注意，剩余的malliavin型抽象的pnt
Note on a conjecture of Bateman and Diamond concerning the abstract PNT with Malliavin-type remainder
论文作者
论文摘要
给定的$β\在（0,1）$中，我们显示了一个beurling通用数字系统的存在，其整数计数满足$ n（x）= ax + o \ bigl（x \ exp（-c \ log^βx x x）\ bigr）$，对于某些$ a> 0 $ a> 0 $ c> 0 $，其prime Counting untun unly unter Counting Function unly且满足$ functies $ $ functies $ $ functies $ functies $ functies $ functies $ funcy（x） ω\ bigl（x \ exp（-c'（\ log x）^{\fracβ{β+1}}}）\ bigr）\ bigr）$ bigr）$）$ c'> 0 $。这是通过概括钻石，蒙哥马利和Vorhauer的结构来完成的。这种Beurling系统是1969年以来对Bateman和Diamond的猜想的额外动机，涉及Malliavin型的剩余时间。
Given $β\in(0,1)$, we show the existence of a Beurling generalized number system whose integer counting satisfies $N(x) = ax + O\bigl(x\exp(-c\log^β x)\bigr)$ for some $a>0$ and $c>0$, and whose prime counting function satisfies $π(x) = \mathrm{Li}(x) + Ω\bigl(x\exp(-c'(\log x)^{\fracβ{β+1}})\bigr)$ for some $c'>0$. This is done by generalizing a construction of Diamond, Montgomery, and Vorhauer. This Beurling system serves as additional motivation for a conjecture of Bateman and Diamond from 1969, concerning the PNT with Malliavin-type remainder.
