Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

For $m,n,c\in\mathbb{N}$ let $S(m,n;c)$ be the Kloosterman sum $$S(m,n;c)=\sum_{a=1, \gcd (a,c)=1}^ce\left(\frac{ma+n\overline{a}}{c}\right).$$ The Kuznetsov Trace Formula allows us to obtain bounds for sums of the form $$\sum_{c\le x,~ c=0\mathrm{mod} q}\frac{S(m,n;c)}{c},$$ which are better than those obtained by simply applying Weil's inequality. I am working on an application for which I need an analogous bound for the sum $$\sum_{p\le x}\frac{S(m,n;p)}{p},$$ where $p$ runs over prime numbers. Does anybody know of a way to obtain a good bound (i.e. better than the Weil bound) for this sum?

share|improve this question
    
I assume $S=K$ above... –  Igor Rivin Nov 25 '11 at 11:10
    
Yes, thank you. –  Alan Haynes Nov 25 '11 at 12:46

1 Answer 1

up vote 5 down vote accepted

No, this is a well-known open problem. One doesn't even know that the sign of $S(1,1;p)$ changes infinitely often... The best that has been achieved are estimates restricted to moudli $c$ with a bounded number of prime factors, by combining sieve methods with automorphic forms (and some average forms of Sato-Tate), see the papers of Fouvry and Michel on the topic.

share|improve this answer
    
There is a survey of related results here: web.science.mq.edu.au/~igor/CharSumProjects.pdf –  Igor Rivin Nov 25 '11 at 11:11
    
That survey is quite interesting, but I don't see anything in it concerning problems of the type raised here (which are sums of Kloosterman sums.) –  Denis Chaperon de Lauzières Nov 25 '11 at 13:01

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.