Upper bounds on character sums with rational function entries (Q1396018)

The authors derive formulae and estimates for general character sums of type \[ S(\chi ,f,p^m) = \sum\limits_{x=1}^{p^m} ~\chi(f(x)), \] where \(p^m\) is a prime power with \(m \geq 2\), \(\chi\) denotes a multiplicative character mod \(p^m\), and \(f=f_1/f_2\) is a rational function over \({\mathbb{Z}}\). In particular, if \(p\) is odd, \(d = \deg(f_1) + \deg(f_2)\) and \(d^* = \max( \deg(f_1), \deg(f_2))\), then \[ | S(\chi, f, p^m)| \leq (d-1) \;p^{m(1- \frac{1}{d^*})} \] for any non-constant \(f\) (mod \(p\)) and primitive character \(\chi\).