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 $p\nmid a$.



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> **Question.** Does anybody know some references concerning sums of this form?
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Character sums concerning $a^x-1$
The best result on the market is the one of [Yu](https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/97/3/82426/estimates-of-character-sums-with-exponential-function), 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](https://www.jstor.org/stable/2159779?seq=1) 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](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=8245&option_lang=eng) 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.


