In this paper we prove: for any positive integers $a$ and $q$ with $\left( a,\,q \right)\,=\,1$, we have uniformly

$$\sum\limits_{\begin{matrix} n\le N \\ (n,q)=1,n\bar{n}\equiv 1(\,\bmod \,q) \\ \end{matrix}}{\mu (n)e(\frac{a\bar{n}}{q})\ll Nd(q)\left\{ \frac{{{\log }^{\frac{5}{2}}}N}{{{q}^{\frac{1}{2}}}}+\frac{{{q}^{\frac{1}{5}}}{{\log }^{\frac{13}{5}}}N}{{{N}^{\frac{1}{5}}}} \right\}.}$$

This improves the previous bound obtained by D. Hajela, A. Pollington and B. Smith [5].

In this article we investigate the average order of the arithmetical function

where p1(t), p2(t) are polynomials in Z [t], of equal degree, positive and increasing for t ≥ 1. Using the modern method for the estimation of exponential sums ("Discrete Hardy-Littlewood Method"), we establish an asymptotic result which is as sharp as the best one known for the classical divisor problem.

An estimate for Kloosterman sums with oscillating coefficients is presented. Precisely we show: for any ϵ > 0 and a, b positive integers with (a, b) — 1 we have,

Similar techniques may be used to estimate other Kloosterman sums with oscillating coefficients which are not smooth.

This paper proves a conjecture of Loxton and Smith about the size of the exponential sum S(f;q) formed by summing exp (2πif(x)/q) over x mod q, where f is a polynomial of degree n with integer coefficients. It is shown that |S(f;q)| ≤ Cfdn(q)qe/(e+1), where e is the maximum of the orders of the complex zeros of f'. An estimate is also obtained for Cf in terms of n, e and the different of f, and a number of examples are given to show that the estimate is best possible.

For suitable polynomials f(x) ∊ ℤ[x] in n variables, of total degree d, it is shown that

This is, formally, a precise analogue of a theorem of Deligne [1] on exponential sums (mod p). However the proof uses no more than elementary algebraic geometry.