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Further estimate of complete trigonometric sums. (English) Zbl 0934.11038
Summary: The following theorem is proved: Let \(f(x)= a_k z^k+\cdots+ a_1x+ a_0\) be a polynomial with integral coefficients such that \((a_1,\dots, a_k,q)=1\), where \(q\) is a positive integer. Then, for \(k\geq 3\), \[ \Biggl|\sum_{x=1}^q e^{2\pi if(x)/q} \Biggr|\leq e^{1.74k} q^{1-\frac 1k}. \] This improved earlier estimates of Nechaev, Chen Jingrun, Stechkin, Lu Minggao and the authors.

MSC:
11L07 Estimates on exponential sums
11L03 Trigonometric and exponential sums (general theory)
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