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An Application of Exponential Sum Estimates

Authors:	Email author" target="_blank">Yuan?Yi Email author Wen?Peng?Zhang

Institution:	(1) Research Center for Basic Science, Xi’an Jiaotong University, Xi’an 710049, P. R. China

Abstract:	Let p be an odd prime and let δ be a fixed real number with 0<δ<2. For an integer a with 0<a<p, denote by ā the unique integer between 0 and p satisfying a ā≡1 (mod p). Further, let {x} denote the fractional part of x. We derive an asymptotic formula for the number of pairs of integers (a, b with $1 \leqslant a \leqslant p - 1,1 \leqslant b \leqslant p - 1,{\left\| {{\left\{ {\frac{{a^{k} }} {p}} \right\}} + {\left\{ {\frac{{b^{k} }} {p}} \right\}} - {\left\{ {\frac{{\overline{a} ^{l} }} {p}} \right\}} - {\left\{ {\frac{{\overline{b} ^{l} }} {p}} \right\}}} \right\|} < \delta$ . This work is supported by N. S. F. of P. R. China (10271093)

Keywords:	Exponential sum Trigonometric sums Asymptotic formula
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