Some estimates of trigonometric sums and their applications
This paper investigates the general classical Kloosterman sum \[ R_{n,m}(u,v) = \sum_{\substack{ 1\leq a,b \leq n\\ ab\equiv 1\pmod n}} e\left({au+bv\over m}\right), \] where \(m,n,u,v\) are integers with \(m\geq 2\), \(n\geq 3\), and \(e(y) = e^{2\pi i y}\). When \(m=n\), this is the familiar Kloosterman sum which satisfies the Weil estimate \(\ll (u,v,n)^{1\over 2}n^{1\over 2}d(v)\), where \((u,v,n)\) denotes the greatest common divisor of \(u\), \(v\), and \(n\). The author shows that if \(m\neq n\), then when \(n\) is a prime one obtains an asymptotic formula for \(R\) with a main term (which vanishes if \(m=n\).) The result is that if \(p\) is an odd prime, and \(m,u,v\) are integers with \(m\nmid u\), \(n\nmid v\), then \[ R_{p,m}(u,v) = {1\over p}e\left({(u+v)(p+1)\over 2m}\right) {\sin{\pi p u\over m} \sin{\pi p v\over m}\over \sin{\pi u\over m}\sin{\pi v\over m}} +O(\sqrt{p}\log ^2p). \] The proof of this result is remarkably simple, with the main term arising naturally and the error terms estimated by the Weil estimate above. By the same method the author next examines the distribution of multiplicative inverses \(\overline{a}\) determined by \(a\overline{a} \equiv 1 \pmod n\) when \((a,n) = 1\). Define \[ S(n,k,\sigma) = \sideset{}{'}\sum_{\substack{ a=1\\ | a- \overline{a}| <\sigma n}}^n | a- \overline{a}| ^k, \] for \(k\geq 0\), \(0\leq \sigma \leq 1\), and \(\sum_a'\) indicates a sum over \(a\) with \((a,n)=1\). The author proves \[ S(n,k,\sigma)= 2\phi(n)n^k \left ({\sigma^{k+1}\over k+1}- {\sigma^{k+2}\over k+2}\right) + O\left(n^{k + {1\over 2}}d^2(n)\log ^2 n\right). \] He also obtains a similar result for \(-1<k<0\). Further, he proves that for \(p\) an odd prime, and \(m\) an integer with \(1<m<p\), then \[ \# \left\{a : 1\leq a\leq p-1, \;a\overline{a}\equiv 1 \pmod p, a\equiv \overline{a} \pmod m\right\} = \tfrac pm+ O(\sqrt{p}\log ^2 p). \] From these results a number of corollaries are obtained for sums involving \(a-\overline{a}\).
scientific article; zbMATH DE number 1244248