Estimates of character sums with exponential function (Q2717597)

Let \(n \geq 2\) and \(\lambda\) be integers satisfying \((n, \lambda)=1\) and \(\lambda\) belonging to the exponent \(d\) modulo \(n\). Given a primitive Dirichlet character \(\chi\) mod \(n\) the author proves the estimate NEWLINE\[NEWLINE \left|\sum_{x=1}^X \chi(a \lambda^x +b) \right|< \sqrt{n} \left( \frac{2}{\pi} \log n + \frac{7}{5} \right). NEWLINE\]NEWLINE Here \(a, b\) and \(X\) are integers satisfying \((ab,n)=1\) and \(1 \leq X \leq d\). In certain cases this inequality is essentially best possible. It is also shown that all results hold in general finite fields.