On the Order of Magnitude of the Divisor Function.

Let $${\fancyscript D}$$ be an increasing sequence of positive integers, and consider the divisor functions: 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 $$ {\sum\nolimits_{n = 1}^\infty {\alpha _{n} d_{2} {\left( {n,{\fancyscript D}} \right)}} } $$ . [ABSTRACT FROM AUTHOR]