Sums of two relatively prime k-th powers

Abstract

Let $k$ be a natural number, $k \ge 3$. Let $V_k(x)$ be the number of solutions $(u, v)$ of \[|u|^k + |v|^k \le x,\qquad (u, v) = 1\] and let \[E_k(x) = V_k(x) - \frac{3\Gamma^2(1/k)}{\pi^2\Gamma(2/k)} x^{2/k}.\] Under the Riemann hypothesis, it is known that \[E_k(x) = O(x^{\t_k+\e})\] for $k = 3, 4$, where $\t_3 = \frac{9581}{36864}$ and $\t_4 = \frac{107}{512}$. The result for $k=3$ is shown to require only a~zero-free strip $Re s > 0.5802\ldots$ for $\z(s)$. For $k=4$, the exponent is improved by about 0.0016. Here the required zero-free strip is $Re s > 0.7058\ldots$. Along the way, new results are obtained for the mean value on a vertical segment $[\s, \s + iT]$ of the Hlawka zeta function defined by \[Z_k(s) = \sum_{n=1}^\infty \frac{r_k(n)}{n^s} \qquad (Re s >1),\] where $r_k(n) = \#\{(x, y) : |x|^k + |y|^k = n\}$. These mean value results improve those in the literature for all $k\ge 3$. The mean value of $E_k(x)$ is also considered, and here an asymptotic formula is provided under the assumption of a zero-free strip of width $> 1/k$. Previous writers required $k \ge 5$ and the Riemann hypothesis.