## Sums of two relatively prime $$k$$-th powers.(English)Zbl 1206.11122

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 $$>\frac{1}{k}$$.

### MSC:

 11P21 Lattice points in specified regions 11D75 Diophantine inequalities 11J25 Diophantine inequalities 11D45 Counting solutions of Diophantine equations

### Keywords:

sums of two $$k$$-th powers
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