Sums of two relatively prime \(k\)-th powers
Let \(k\geq 3\) be a natural number. Let \(V_k(x)\) be the number of solutions \((u,v)\) in \({\mathbb Z}^2\) of \[ |u|^k+|v|^k\leq x,\quad (u,v)=1 \] and let \[ E_k(x)=V_k(x)-c_kx^{\frac{2}{k}},\quad c_k=\frac{3}{\pi^2}\frac{\Gamma^2(\frac{1}{k})}{\Gamma(\frac{2}{k})}. \] The author proves: Theorem 1. If \(\zeta(s)\) has no zero with real part greater than \(\frac{123\theta_3-30}{90\theta_3-20}\), \(\theta_3=\frac{9581}{36864}\) then for every \(\varepsilon>0\) \[ E_3(x)=O(x^{\theta_3+\varepsilon}). \] Theorem 2. If \(\zeta(s)\) has no zero with real part greater than \(\frac{32\theta_4-5}{16\theta_4-1}\), \(\theta_4=\frac{7801}{37616}\) then for every \(\epsilon>0\) \[ E_4(x)=O(x^{\theta_4+\varepsilon}). \] 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 \(>\frac{1}{k}\).
scientific article; zbMATH DE number 5693244