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I am interested in an upper bound on the following incomplete Kloosterman sum $$ \sum_{\substack{x \text{ mod }p \\ x+x^{-1}>p}}e\left(\frac{x+x^{-1}}{p}\right).$$

Using the Weil's bound it is easy to show that the real part of the sum is bounded by $\sqrt p.$ It is because if $x+x^{-1}>p$ then $(p-x) + (p-x)^{-1}< p.$ Note that $xx^{-1} \equiv 1 \text{ mod } p.$

I am interested in an upper bound on the following incomplete Kloosterman sum $$ \sum_{\substack{x=1 \\ x+_{_{\bf Z}}x^{-1}>p}}^{p-1}e\left(\frac{x+x^{-1}}{p}\right).$$

Using the Weil's bound it is easy to show that the real part of the sum is bounded by $\sqrt p.$ It is because if $x+x^{-1}>p$ then $(p-x) + (p-x)^{-1}< p.$ Note that $x^{-1} \in \{1, \cdots , p-1\}$ and $xx^{-1} \equiv 1 \text{ mod } p$. And by $+_{_{\bf Z}}$ we mean that the sum in $\bf Z$ not in $\bf Z_p$.

I am interested in an upper bound on the following incomplete Kloosterman sum $$ \sum_{\substack{x \text{ mod }p \\ x+x^{-1}>p}}e\big(\frac{x+x^{-1}}{p}\big).$$

Using the Weil's bound it is easy to show that the real part of the sum is bounded by $\sqrt p.$ It is because if $x+x^{-1}>p$ then $(p-x) + (p-x)^{-1}< p.$ Note that $xx^{-1} \equiv 1 \text{ mod } p.$

I am interested in an upper bound on the following incomplete Kloosterman sum $$ \sum_{\substack{x \text{ mod }p \\ x+x^{-1}>p}}e\left(\frac{x+x^{-1}}{p}\right).$$

Using the Weil's bound it is easy to show that the real part of the sum is bounded by $\sqrt p.$ It is because if $x+x^{-1}>p$ then $(p-x) + (p-x)^{-1}< p.$ Note that $xx^{-1} \equiv 1 \text{ mod } p.$