Modular Equations and Approximations to π

Abstract

If we suppose that

$$(1 + {e^{ - \pi Nn}})(1 + {e^{ - 3\pi Nn}})(1 + {e^{ - \beta \pi Nn}})... = {2^t}{e^{ - \pi Nn/24}}{G_n}......... $$

((1))

and

$$(1 - {e^{ - \pi Nn}})(1 - {e^{ - 3\pi Nn}})(1 - {e^{ - \beta \pi Nn}})... = {2^t}{e^{ - \pi Nn/24}}{g_{n,}}........$$

((2))

then G _n and g _n can always be expressed as roots of algebraical equations when n is any rational number.