Bounds of incomplete multiple Kloosterman sums
Let \(p\) be a prime number. For an integer \(n\) not divisible by \(p\) let \(\bar{n}\) be the inverse of \(n\) modulo \(p\): \(n\bar{n}\equiv 1 (\bmod p),\) \(1\leq \bar{n}<p.\) Let \({\mathbf e}_p(z)=\exp(\frac{2\pi i z}{p} )\). The author considers incomplete \(s\)-dimensional Kloosterman sums \[ K_s( {\mathbf a},{\mathbf M},{\mathbf N};p )= \sum_{n_1=M_1+1}^{M_1+N_1}\ldots\sum_{n_s=M_s+1}^{M_s+N_s}{\mathbf e}_p(a_1n_1+\ldots+a_sn_s+a_{s+1}\overline{n_1\ldots n_s}), \] with integer vectors \( {\mathbf a}=(a_1,\ldots,a_{s+1})\in {\mathbb Z}^{s+1}\) and \( {\mathbf N}=(N_1,\ldots,N_s)\), \( {\mathbf M}=(M_1,\ldots,M_s)\in {\mathbb Z}^s\) such that \(0 \leq M_\nu<M_\nu+N_\nu<p\), \(\nu=1,\ldots,s\). He proves: Theorem 1. For any integer \(s\geq 2\), uniformly over all integer vectors \( {\mathbf a}=(a_1,\ldots,a_{s+1})\in {\mathbb Z}^{s+1}\), \({\mathbf N}=(N_1,\ldots,N_s)\), \( {\mathbf M}=(M_1,\ldots,M_s)\in {\mathbb Z}^s\) such that \(0 \leq M_\nu<M_\nu+N_\nu<p\), \(\nu=1,\ldots,s\), and \(\gcd(a_{s+1}, p)=1\), and any integer \(r\geq 2\), we have \[ K_s( {\mathbf a},{\mathbf M},{\mathbf N};p )\ll N_1 \ldots N_s p^{-1}+(N_1 \ldots N_s)^{1-(r+s-2)/rs}p^{1/2+(s-2)/4(r-1)}(\log p)^{2s-4}. \] Theorem 2. For any integer \(s\geq 4\), uniformly over all integer vectors \( {\mathbf a}=(a_1,\ldots,a_{s+1})\in {\mathbb Z}^{s+1}\), \({\mathbf N}=(N_1,\ldots,N_s)\), \( {\mathbf M}=(M_1,\ldots,M_s)\in {\mathbb Z}^s\) such that \(0 \leq M_\nu<M_\nu+N_\nu<p\), \(\nu=1,\ldots,s\), and \(\gcd(a_{s+1}, p)=1\), and any integer \(r\geq 2\), we have \[ K_s( {\mathbf a},{\mathbf M},{\mathbf N};p )\ll N_1 \ldots N_s p^{-1}+(N_1 \ldots N_s)^{1-(s-4)/rs}p^{-1/2+(s-4)/4(r-1)}W(\log p)^{2s-4}, \] where \[ W=\prod_{\nu=1}^s(1+p^{1/s}N_\nu^{-2/s}(\log p)^{2/s}). \]
scientific article; zbMATH DE number 5190932