For any positive integer $k\ge 3$, it is easy to prove that the $k$-polygonal numbers are $a_n\left(k\right)=(2n+n(n-1)(k-2))/2$. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet $L$-functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums $S(a_n\left(k\right){\overline{a}}_m\left(k\right),p)$ for $k$-polygonal numbers with $1\le m,n\le p-1$, and give an interesting computational formula for it.

The $L^1$ norm of a trigonometric polynomial with $N$ non zero coefficients of absolute value not less than 1 exceeds a fixed positive multiple of $C\left(\log N\right)/(\log \log N{)}^2.$

The main purpose of this paper is to study the mean value properties of a sum analogous to character sums over short intervals by using the mean value theorems for the Dirichlet L-functions, and to give some interesting asymptotic formulae.

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...

The main purpose of the paper is to study, using the analytic method and the property of the Ramanujan’s sum, the computational problem of the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum. For integers $m$, $n$, $k$, $q$, with $k\ge 1$ and $q\ge 3$, and Dirichlet characters $\chi$, $\overline{\chi }$ modulo $q$ we define a mixed exponential sum $$C(m,n;k;\chi ;\overline{\chi };q)=\sum _{a=1}^q{width0ptheight1em}^{\text{'}}\chi \left(a\right)G_k(a,\overline{\chi })e\left(\frac{m{a}^k+n\overline{a^k}}{q}\right),$$ with Dirichlet character $\chi$ and general Gauss sum $G_k(a,\overline{\chi })$ as coefficient, where ${\sum }^{\text{'}}$ denotes the summation over all $a$ such that $(a,q)=1$, $a\overline{a}\equiv 1modq$ and $e\left(y\right)={\mathrm{e}}^{2\pi \mathrm{i}y}$. We mean value of $$\sum _m\sum _{\chi }\sum _{\overline{\chi }}{\left|C(m,n;k;\chi ;\overline{\chi };q)\right|}^4,$$ and...

Let $k$ be a fixed integer. We study the asymptotic formula of $R(H,r,k)$, which is the number of positive integer solutions $1\le x,y,z\le H$ such that the polynomial $x^2+y^2+z^2+k$ is $r$-free. We obtained the asymptotic formula of $R(H,r,k)$ for all $r\ge 2$. Our result is new even in the case $r=2$. We proved that $R(H,2,k)=c_k{H}^3+O\left(H^{9/4+\epsilon }\right)$, where $c_k>0$ is a constant depending on $k$. This improves upon the error term $O\left(H^{7/3+\epsilon }\right)$ obtained by G.-L. Zhou, Y. Ding (2022).