We consider the Legendre quadratic forms$${\varphi }_{ab}(x,y,z)=a{x}^2+b{y}^2-z^2$$and, in particular, a question posed by J–P. Serre, to count the number of pairs of integers $1\le a\le A,1\le b\le B$, for which the form ${\varphi }_{ab}$ has a non-trivial rational zero. Under certain mild conditions on the integers $a,b$, we are able to find the asymptotic formula for the number of such forms.

Let $q$ be a positive integer, $\chi$ denote any Dirichlet character $modq$. For any integer $m$ with $(m,q)=1$, we define a sum $C(\chi ,k,m;q)$ analogous to high-dimensional Kloosterman sums as follows: $$C(\chi ,k,m;q)=\sum _{a_1=1}^q{}^{\text{'}}\sum _{a_2=1}^q{}^{\text{'}}\cdots \sum _{a_k=1}^q{}^{\text{'}}\chi (a_1+a_2+\cdots +a_k+m\overline{a_1{a}_2\cdots a_k}),$$ where $a·\overline{a}\equiv 1modq$. The main purpose of this paper is to use elementary methods and properties of Gauss sums to study the computational problem of the absolute value $\left|C\right(\chi ,k,m;q\left)\right|$, and give two interesting identities for it.