Exponential sums modulo prime powers (Q2773326)

The author develops a method for evaluating and estimating a very general mixed exponential sum of type NEWLINE\[NEWLINE S( \chi, g, f, p^m)= \sum_{k=1}^{p^m} \chi (g(k)) e_{p^m} (f(k)), NEWLINE\]NEWLINE where \(p^m\) is a prime power with \(m \geq 2\), \(\chi\) is a multiplicative character \(\operatorname {mod}p^m\), \(e_{p^m}(x) = e^{2 \pi i x/p^m}\), and \(f,g\) are rational functions with integer coefficients. The sum is only over values of \(k\) for which \(g\) and \(f\) are both defined as functions on \({\mathbb{Z}}/(p^m)\), and \(g\) is nonzero \(\operatorname {mod}p\) . The proof uses the \(p\)-adic logarithm to describe the behavior of \(\chi\) on the subgroup of residues \(\operatorname {mod}p^m\) congruent to \(1\) \(\operatorname {mod}p\).