Effective and noneffective results on certain arithmetical functions
Let \(a\geq 2\) be an integer, \(l,m\) be complex numbers, \(\alpha=\tfrac12-\tfrac12 a\), \[ F(s)= (\xi(s))^a (\xi(2s))^l (\xi(3s))^m = \sum_{n=1}^\infty a_n n^{-s}, \] \[ M(x) = \sum \text{Re}\,s\{F(s)x^s/s\},\quad N(x) = \sum_{n\leq x} a_n,\quad E(x) = N(x)-M(x). \] It is proved that there exists an effective constant \(X_0 = X_0(a,l,m,\varepsilon)\) such that for \(X\geq X_0\) \[ \int_X^{(a+1+\varepsilon)/2} | E(x)x^{-\alpha}|^2 \,dx/x > X^{-\varepsilon}. \] In particular, this result implies that, if \(\mathcal D(n)\) is the number of square free divisors of \(n\), \[ E(x)= \sum_{n\leq x} \mathcal D(n) - x/\zeta(2)(\log x +2\gamma - 1 - 2\zeta'(2)/\zeta(2)), \] then for \(x\geq x_0\) \[ \max_{X\leq x\leq X^{(3+\varepsilon)/2}} | E(x)/x^{1/4 - \varepsilon}| \geq 1. \]
scientific article; zbMATH DE number 3666897
scientific article