On twisted character sums
Let \(p\) be a prime, and let \(\chi\) be a non-principal character modulo \(p\). Let \(f\) be a smooth \(1\)-periodic function and \(\alpha\in \mathbb{R}\), and let \(M,N,r\) be positive integers with \(M,N<p\) and \(r>1\). This paper proved that \[ \sum_{M<n\leq M+N}f(\alpha n)\chi(n)\ll ||f||_{2,1} N^{1-1/r}p^{1/(4r-4)}(\log p)^{1/2r}, \] where \(||f||_{2,1}=\int_{0}^{1}\left(|f|+|f''|\right)\) and the \(\ll\)-constant is absolute.
scientific article; zbMATH DE number 5909099
scientific article