Let $D$ be a positive integer, and let $p$ be an odd prime with $p\nmid D$. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M. A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for $N(D, p)$, and also prove that if the equation $U^2-DV^2=-1$ has integer solutions $(U, V)$, the least solution $(u_1, v_1)$ of the equation $u^2-pv^2=1$ satisfies $p\nmid v_1$, and $D>C(p)$, where $C(p)$ is an effectively computable constant only depending on $p$, then the equation $x^2-D=p^n$ has at most two positive integer solutions $(x, n)$. In particular, we have $C(3)=10^7$.
The various properties of classical Dedekind sums $S(h, q)$ have been investigated by many authors. For example, Yanni Liu and Wenpeng Zhang: A hybrid mean value related to the Dedekind sums and Kloosterman sums, Acta Mathematica Sinica, 27 (2011), 435–440 studied the hybrid mean value properties involving Dedekind sums and generalized Kloosterman sums $K(m, n, r; q)$. The main purpose of this paper, is using the analytic methods and the properties of character sums, to study the computational problem of one kind of hybrid mean value involving Dedekind sums and generalized Kloosterman sums, and give an interesting identity.
Various properties of classical Dedekind sums $S(h, q)$ have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H. Rademacher and E. Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to study the computational problem of one kind of mean value involving the classical Dedekind sum and the quadratic Gauss sum, and give several exact computational formulae for it.
For the general modulo $q\geq 3$ and a general multiplicative character $\chi $ modulo $q$, the upper bound estimate of $ |S(m, n, 1, \chi , q)| $ is a very complex and difficult problem. In most cases, the Weil type bound for $ |S(m, n, 1, \chi , q)| $ is valid, but there are some counterexamples. Although the value distribution of $ |S(m, n, 1, \chi , q)| $ is very complicated, it also exhibits many good distribution properties in some number theory problems. The main purpose of this paper is using the estimate for $k$-th Kloosterman sums and analytic method to study the asymptotic properties of the mean square value of Dirichlet $L$-functions weighted by Kloosterman sums, and give an interesting mean value formula for it, which extends the result in reference of W. Zhang, Y. Yi, X. He: On the $2k$-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums, Journal of Number Theory, 84 (2000), 199–213.