On differences of two \(k\)th powers: An asymptotic formula for the mean-square of the error term
Let \(k\geq 3\) be a fixed natural number and \(r_k^-(n)\) be the arithmetic function which counts the number of representations of the positive integer \(n\) as a difference of \(k\)th powers of two integers taken absolutely: \[ r_k^-(n)= \#\{(n_1,n_2)\in \mathbb{Z}^2: |n_1|^k-|n_k|^k= n\}. \] The reviewer [Acta Arith. 16, 111-121 (1969; Zbl 0201.05003)] established an asymptotic formula for the sum \[ T_k^-(x)= \sum_{1\leq n\leq x} r_k^-(n). \] It is \[ T_k^-(t^k)= a_kt^2+ b_kt^{k/(k-1)}+ c_k(t) t^{1-1/k}+ \Delta_k^-(t), \] where \(a_k, b_k\) are some constants and \(c_k(t)\) a well-defined function of order 1. \textit{W. G. Nowak} [Analysis 16, 297-304 (1996; Zbl 0860.11060)] proved the estimation \[ \Delta_k^-(t)= O(t^{23/73} (\log t)^{315/146}) \] and [Ramanujan J. 2, 421-440 (1998; Zbl 0922.11080)] \[ \Delta_k^-(t)= \Omega_+ (t^{1/2} (\log t)^{1/2k} \log\log t), \qquad \int_0^x (\Delta_k^-(t))^2 dt= O(x^2). \] The last result is now improved by the author to \[ \lim_{x\to\infty} \frac{1}{x^2} \int_0^x (\Delta_k^-(t))^2 dt= C_k>0. \] The proof is difficult, because the corresponding Dirichlet series to the arithmetic function \(r_k^-(n)\) does not satisfy a functional equation.
scientific article; zbMATH DE number 1310082