Bounds on bilinear forms with Kloosterman sums
Bounds on bilinear forms with Kloosterman sums
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DOI：
10.1112/jlms.12753
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发表时间：
2022-04
期刊：
影响因子：
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通讯作者：
Bryce Kerr;I. Shparlinski;Xiaosheng Wu;Ping Xi
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文献类型：
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作者：
Bryce Kerr;I. Shparlinski;Xiaosheng Wu;Ping Xi
我们用Kloosterman和证明了双线性形式的新界，通过É补充和改进了一系列结果。Fouvry， E. Kowalski和Ph. Michel (2014)， V. Blomer， É。Fouvry， E. Kowalski, Ph. Michel and D. Milićević （2017）， E. Kowalski, Ph. Michel and W. Sawin（2019,2020）和I. Shparlinski（2019）。这些改进依赖于对不完全Kloosterman和的II型双线性形式的新估计。通过引入素域上的加性组合技术，我们建立了任意集单变量双线性形式的新估计。在Wu（2020）最近关于Dirichlet L$L$‐函数的第四阶矩渐近公式的工作中，这些界限中的一些已经找到了重要的应用。作为新的应用，给出了一类等差数列中Kloosterman和的高平均矩的估计和除数函数的分布。
We prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by É. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, É. Fouvry, E. Kowalski, Ph. Michel and D. Milićević (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new estimates for Type II bilinear forms with incomplete Kloosterman sums. We also establish new estimates for bilinear forms with one variable from an arbitrary set by introducing techniques from additive combinatorics over prime fields. Some of these bounds have found a crucial application in the recent work of Wu (2020) on asymptotic formulas for the fourth moments of Dirichlet L$L$ ‐functions. As new applications, an estimate for higher moments of averages of Kloosterman sums and the distribution of divisor function in a family of arithmetic progressions are also given.