On the Order of Magnitude of the Divisor Function

Abstract

Let ${\fancyscript D}$ be an increasing sequence of positive integers, and consider the divisor functions:

$$ \begin{array}{*{20}c} {{d{\left( {n,{\fancyscript D}} \right)} = {\sum\limits_{\begin{array}{*{20}c} {{d\left| n \right.}} \\ {{d \in {\fancyscript D},d \leqslant {\sqrt n }}} \\ \end{array} } {1,} }}} & {{d_{2} {\left( {n,{\fancyscript D}} \right)} = {\sum\limits_{\begin{array}{*{20}c} {{{\left[ {d,\delta } \right]}\left| n \right.}} \\ {{d,\delta \in {\fancyscript D},{\left[ {d,\delta } \right]} \leqslant {\sqrt n }}} \\ \end{array} } {1,} }}} \\ \end{array} $$

where [d, δ] = l.c.m.(d, δ). A probabilistic argument is introduced to evaluate the series ${\sum\nolimits_{n = 1}^\infty {\alpha _{n} d{\left( {n,{\fancyscript D}} \right)}} }$ and ${\sum\nolimits_{n = 1}^\infty {\alpha _{n} d_{2} {\left( {n,{\fancyscript D}} \right)}} }$.