Prime-power Gauss sum bound
= Prime-power Gauss sum bound
{title2=$|q^{-1/2}\tau(\chi)|\le1$}
For a character of modulus $p^k$, a <primitive Dirichlet character> attains the bound by finite <Plancherel theorem>. An imprimitive character with $k\ge2$ has periodic values modulo $p^{k-1}$, so its frequency-one sum vanishes. The <principal Dirichlet character> modulo $p$ has unnormalized sum $-1$. The sign convention of the exponential does not affect its magnitude.