## Salie-type sum bound

I am interested in bounding the following Salie-type ("twisted Kloosterman") sum

$$S(a,b,\beta) = \sum_{x \in \mathbb{Z}/{p^{\beta}}\mathbb{Z}} \left( \frac{x}{p^{\beta}} \right) \chi(ax + bx^{-1}).$$

Here, $\left( \frac{\cdot}{q} \right)$ denotes the Jacobi symbol, $\chi(x) = \exp(2 \pi i x /p^{\beta})$, $p$ is an odd prime, $\beta$ is a positive integer, and I am always assuming $\gcd(p,ab) = 1$.

I am trying to find a bound for these Salie-type sums. When $\beta = 1$, the bound $|S(a,b,1)| \leq 2 \sqrt{p}$ is due to Salie. When $\beta \geq 2$ is even, the Jacobi symbol is identically $1$ and so the sum reduces to a Kloosterman sum

$$K(a,b,\beta) = \sum_{x \in U(\mathbb{Z}/{p^{\beta}} \mathbb{Z})} \chi(ax + bx^{-1})$$

and the well known Weil bound applies giving $|K(a,b,\beta)| \leq (\beta + 1) p^{\beta/2}$. Here, $U(R)$ denote the set of units in $R$.

My question is: How does one bound the sum $S(a,b,\beta)$ in the case $\beta \geq 3$ is odd? It seems reasonable that the bound $|S(a,b,\beta)| \leq (\beta + 1) p^{\beta/2}$ should apply here as well.

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There is a general "elementary" formula for Salié sums for arbitrary modulus, involving roots of quadratic equations, and from which the bound is immediate. A quick derivation is in Sarnak's "Some applications of modular forms" but it can be found in many places.

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Another nice reference is H. Iwaniec's Topics in classical automorphic forms, in which he computed the classical Kloosterman sums and Salie sums in an elementary manner for the prime power moduli. – arithboy May 19 2011 at 6:44
In fact, a similar upper bound also holds if the Jacobi symbol is replaced by any other Dirichlet character mod $p^\beta$. – arithboy May 19 2011 at 6:46
@Denis @arithboy: Thanks for the references! – David May 19 2011 at 14:45

I beleive this is done in Keith Conrad's paper, On Weil's proof of the bound for Kloosterman sums, J. Number Theory 97 (2002), no. 2, 439–446, MR1942969 (2003j:11087).

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 Why did this get voted up? That paper does not address the question that is asked, as it only treats the case of sums over finite fields, not finite rings as in the question (with beta > 1). – – KConrad May 19 2011 at 6:51 Apologies to all, misreading on my part. – Gerry Myerson May 19 2011 at 10:31