Character sums concerning $a^x-1$ Let $p$ be a prime number and $\mathbb{F}_p$ be a finite field with $p$ elements. Let $\chi$ be a multiplicative character from $\mathbb{F}_p^{\times}$ to $\mathbb{C}$, where $\mathbb{F}_p^{\times}=\{x\in\mathbb{F}_p: x\ne 0\}$.
Recently, I met the following sum:
$$\sum_{x=0}^{p-1}\chi(a^x-1),$$
where $a>1$ is a positive integer with $p\nmid a$.

Question. Does anybody know some references concerning sums of this form?

 A: The best result on the market is the one of Yu, who proves that your sum is less than $$p^{1/2} \left( \frac 2 \pi \log p + \frac 7 5 \right).$$ If $a$ is not a primitive root there is an adjustment by a factor which depends on the index of $a$. Also see Dobrowolski&Williams for essentially the same bound in the special case of Legendre symbols. Yu also proves the same bound for prime power moduli. See moreover Shparlinski for non-primitive characters, where there is an extra factor depending on the conductor of $\chi$; again from Yu, the displayed bound is in this case best possible up to a $\log q$ factor. Shparlinski and others comment that much less is studied for this kind of sums, involving multiplicative characters, than for the analogues with additive characters, especially for short sums.
A: Shparlinski has done a lot of work on problems like this, in the more general setting of linear recurrence sequences. (Your sequence of Mersenne numbers is such a sequence).
I'd recommend looking at Chapter 5 of
Graham Everest, Alf van der Poorten, Igor Shparlinski, Thomas Ward - Recurrence Sequences
and the references therein (see e.g. p 86).
