Selberg筛权重与Kloosterman和的符号变化. I. 平移权重的均匀渐近
Selberg sieve weights and sign changes of Kloosterman sums. I. Uniform asymptotics for shifted weights
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中文总结 AI 辅助
本文证明带二次截断函数的除数和关联的均匀渐近公式，误差为O_g(Y/(log Y)^2)，主项为导数双线性形式，为Kloosterman和符号变化问题提供关键估计。
中文摘要 AI 辅助
我们证明了带有二次截断函数的除数和的关联的渐近公式，其中截断函数分别独立地平移了\\(s,u\in[0,1/2]\\)。误差项为\\(O_g(Y/(\log Y)^2)\\)，对\\(Y^{1/5}\leq R\leq Y^{1/3}\\)一致成立，包括重合平移的情形。主项是截断函数的一阶和二阶导数的显式双线性形式。我们通过在外层Mellin变量中取留数，并利用Laplace反演计算剩余的二重积分来得到该结果。本文建立了除数和估计。其在符号变化问题中的应用，包括素数截断的选择和其余参数的选取，将在第二部分中处理。
英文摘要
We prove an asymptotic formula for correlations of divisor sums with quadratic cutoff functions shifted independently by \(s,u\in[0,1/2]\). The error is \(O_g(Y/(\log Y)^2)\), uniformly for \(Y^{1/5}\leq R\leq Y^{1/3}\), including coincident shifts. The main term is an explicit bilinear form in the first and second derivatives of the cutoff functions. We obtain it by taking the residue in the outer Mellin variable and evaluating the remaining double integral by Laplace inversion. The present paper establishes the divisor-sum estimate. Its application to the sign-change problem, including the choice of the prime cutoff and the remaining parameters, is treated in Part II.
发表机构
- School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
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