About a theorem of Paolo Codecà's and \(\Omega\)-estimates for arithmetical convolutions
Let \(\phi\) and \(\sigma\) denote Euler's totient and the sum of divisor functions, respectively, and \[ H(x)=\sum_{n\leq x}\frac{\phi (n)}{n}- \frac{6x}{\pi^ 2},\quad G(x)=\sum_{n\leq x}\frac{n}{\phi (n)}- \frac{315 \zeta (3)}{2\pi^ 4}x+ Log x \] \[ and\quad F(x)=\sum_{n\leq x}\frac{\sigma (n)}{n}-\frac{\pi^ 2}{6}x+ Log x, \] \(\zeta(s)\) being the Riemann zeta-function. In this paper, the author proves a general theorem which can be used to obtain various Omega results. In particular, he proves that \(F(x)=\Omega_{\pm}(Log Log x)\) and \(G(x)=\Omega_{\pm}(Log Log x)\) sharpening earlier results due to P. Codecà, the author and the reviewer. As another application, he proves that if \(N_ H(x)\) denotes the number of sign changes of H(n), \(n\leq x\), then \[ N_ H(x)\geq (Log Log x)^{3/2}\quad (Log Log Log x)^{-2}. \] This sharpens an earlier result due to the author [Acta Arith. 51, No.4, 321-333 (1988; Zbl 0613.10038)] while P. Erdős conjectured (in a letter to J. Steinig, 1967) that \(N_ H(x)\sim cx\).
scientific article; zbMATH DE number 4059509