Sharp bounds for summatory functions

Overview & Synopsis

textbf{Abstract. }Let\textbf{\ }$\Omega (n)$ denote the number of prime powers of $n$ counted with multiplicity and let $\sum\nolimits^{\prime }$ denote the summation over odd positive integers. In this paper, we prove that% \begin{equation*} \sum_{n\leq x}^{\prime }\frac{2^{\Omega (n)}}{n}=\frac{C}{2}\log ^{2}x+C_{1}\log x+C_{2}+R(x), \end{equation*}% with \ $\left\vert R(x)\right\vert \leq \frac{C_{3}}{x^{1-a}}$, where $% C_{1},C_{2},C_{3}$\ are computable constants and $\sigma =\log (2)/\log (3)$.