论文标题
对乘法亚组的指数总和的新估计和素数间隔
New estimates for exponential sums over multiplicative subgroups and intervals in prime fields
论文作者
论文摘要
令$ {\ Mathcal H} $为$ \ mathbb {f} _p^*$的乘法子组，$ h> p^{1/4} $。我们表明$$ \ max _ {（a，p）= 1} \ left | \ sum_ {x \ in {\ mathcal h}}} {\ mathbf {\，\，e}} _ p（ax）_ p（ax）\ right | \ le h^{1-31/2880+o（1）}，$$其中$ {\ mathbf {\，e}} _ p（z）= \ exp（2πiz/p）$，这改善了Bourgain and Garaev（2009）的结果。我们还获得了带有product $ nx $的双重指数和$ x \ in {\ mathcal h} $的新估算值，而在{\ Mathcal n} $中，对于连续整数的简短间隔$ {\ Mathcal n} $，{\ Mathcal n} $ in {\ Mathcal n} $。
Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) = \exp(2 πi z/p)$, which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double exponential sums with product $nx$ with $x \in {\mathcal H}$ and $n \in {\mathcal N}$ for a short interval ${\mathcal N}$ of consecutive integers.
