# Sums over the positive integers

Can anyone see why it is that if $a$ is large, then $$\log (\sum_m\sum_n \exp(-knm/a)))$$ where $k$ is a constant and $n,m$ take values $1,2,3,...$, can be approximated by $${a\pi^2\over 6k }$$? Cheers!

As Gerry suggested, it would probably help if I added more context. Let $$Z= \sum_m\sum_n \exp(-{nm\hbar \omega\over k_B T}))$$ and the free energy is $$A= -k_BT \log Z$$ and for this $Z$, and at large  $T$, $A$ is said to be approximately $$-k_b^2T^2\pi^2\over 6\hbar \omega$$

-
Source?${}{}{}$ –  Gerry Myerson Feb 8 at 11:31
@GerryMyerson: it's from some notes I have... –  user61472 Feb 8 at 11:38
@user61472 Well, at least $\sum_n e^{-knm/a} = 1/(e^{k m/a}-1)$ –  Valtteri Feb 8 at 12:13