# Prime character sums

Let $$p$$ be a (large) prime number, and let $$\chi : (\mathbf{Z}/p\mathbf{Z})^{\times} \rightarrow \mathbf{C}^{\times}$$ be a Dirichlet character of conductor $$p$$. We have good estimates on the character sum $$\sum_{n \leq x} \chi(n)$$. For instance, if $$p>10^7$$ then for any $$x\geq 1$$ we have: $$\mid \sum_{n\leq x } \chi(n) \mid \leq 2.74\cdot p^{\frac{1}{2}}\cdot \log(p) \text{ .}$$

I'm looking for similar estimates for the character sum restricted to prime numbers: do we have similar upper-bound for $$\mid \sum_{q\leq x\atop q \text{ prime} } \chi(q) \mid \text{ ?}$$ I would be happy to make some restrictions on $$x$$ (for instance, $$x = O(p^{1-\epsilon})$$ for some $$\epsilon>0$$, or $$x=C\cdot p$$ for some $$0; I want $$x$$ to be large). I really need an $$\textbf{explicit}$$ numerical bound (so all the constants should be effective and relatively small). I would of course prefer something unconditional (on GRH).

My motivation is the following. Suppose $$\ell$$ is a prime dividing $$p-1$$. I want to find an explicit upper-bound on the number of $$\textbf{primes}$$ less than $$x$$ which are $$\ell$$th power modulo $$p$$. If $$\ell$$ is big (much bigger than $$\log(p)$$ for instance) this is not very challenging since the obvious bound $$\frac{p-1}{\ell}$$ becomes good.

• In the book Analytic Number Theory by Iwaniec and Kowalski it is proved (Corollary 5.29) that $$\sum_{q\le x}\chi(q)\ll\sqrt{p} x(\log x)^{-A}$$ for any $A>0$ where the sum is over primes $q$ but this is worse than the trivial bound $x / \log x$ in the range this question is considering. – kodlu Mar 4 at 17:41