一类指数和的加权均值

The weighted mean problem of a class of exponential sums $S(m,n)=\sum_{x = 1}^t e_p(mg^x)e_t(nx)$ has been studied, in the forms of $\sum_{m\in\mathcal{I}}\sum_{n\in\mathcal{J}}\alpha_m S(m,n)$, $\sum_{m\in\mathcal{I}}\sum_{n\in\mathcal{J}}\alpha_m\beta_n S(m,n)$. By using elementary and analytic methods, stronger upper bound estimates for them have been given. The results show that there is a certain cancellation property for this weighted sum.

研究了一类指数和 $S\left( {m,n} \right) = \sum\limits_{x = 1}^t {{e_p}\left( {m{g^x}} \right){e_t}\left( {nx} \right)} $的加权均值问题,形如 $\sum\limits_{m \in \mathcal{I}} {\sum\limits_{n \in \mathcal{J}} {{\alpha _m}S\left( {m,n} \right)} } $, $\sum\limits_{m \in \mathcal{I}} {\sum\limits_{n \in \mathcal{J}} {{\alpha _m}{\beta _n}S\left( {m,n} \right)} } $。利用初等方法和解析方法给出了它们的较强上界估计,结果表明该加权和存在一定的相消性。