= Mass-to-light ratio of a power-law stellar population
Write $x=m/M_\odot$, the <initial mass function> as $dN=A x^{-\alpha}dx$, and a main-sequence <mass-luminosity relation> as $L/L_\odot=x^\beta$. On $a<x<b$, the <mass-to-light ratio> in solar units is
$$
\frac{\Upsilon}{M_\odot/L_\odot}=\frac{\int_a^b x^{1-\alpha}dx}{\int_a^b x^{\beta-\alpha}dx}.
$$
For $\alpha=\beta=3$ this is exactly $1/(ab)$. The result concerns the specified unevolved <stellar population>; <stellar remnants>, evolved luminous stars and cooling substellar objects require different accounting.
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