The Selberg–Linnik bound for twisted Kloosterman sums

Let $N$ be the level, let $m,n\neq0$, and define the twisted partial sum

$$Z^*_{m,n}(x)=\sum_{N\mid c\leq x}\frac{S^*(m,n;c)}{c}.$$

Selberg–Linnik bound. For every $m,n\neq0$ and every $\varepsilon>0$,

$$Z^*_{m,n}(x)\ll_{m,n,\varepsilon}x^\varepsilon.$$

This conjectural bound controls the partial sums at the critical point $s=1/2$ and is intended to describe cancellation in twisted Kloosterman sums; its status is open in the supplied source.