For the general modulo $q\ge 3$ and a general multiplicative character $\chi$ modulo $q$, the upper bound estimate of $\left|S\right(m,n,1,\chi ,q\left)\right|$ is a very complex and difficult problem. In most cases, the Weil type bound for $\left|S\right(m,n,1,\chi ,q\left)\right|$ is valid, but there are some counterexamples. Although the value distribution of $\left|S\right(m,n,1,\chi ,q\left)\right|$ 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...

We examine an arithmetical function defined by recursion relations on the sequence ${\left\{f\left(p^k\right)\right\}}_{k\in \mathbb{N}}$ and obtain sufficient condition(s) for the sequence to change sign infinitely often. As an application we give criteria for infinitely many sign changes of Chebyshev polynomials and that of sequence formed by the Fourier coefficients of a cusp form.