Correlation and lower bounds of arithmetic expressions (Q6595586)

The author of the paper derives some lower bounds for arithmetic quantities. Below is one of the main inequalities proved in the article.\N\NLet \(\mathcal{N}(R)\) be number of elements \(\mathbf{x}=(x,y,z)\in\mathbb{Z}^3\setminus\{0\}\) with coprime coordinates such that \(\| \mathbf{x}\|\leqslant R\). Then, for any \(p>1\) it holds that\N\[\N\int_R^{2R}|\mathcal{E}|^p\gg R^{\,p+1}(\log R)^{p/2},\N\]\Nwhere \(\mathcal{E}=\mathcal{N}(R)-\frac{4\pi}{3\zeta(3)}N^3\).