A double Stirling formula

Abstract:

We give an asymptotic expansion as $M \to \infty ,\;N \to \infty$ of $\prod {{{(m + n\tau )}^2}}$, where the product is over $1 \leqslant |m| \leqslant M,\;1 \leqslant |n| \leqslant N,(m,n) \ne (0,0)$. The formula is analogous to the classical Stirling expansion on $M!$. Of special interest is the constant term in the expansion, which involves the Dedekind eta function $\eta (z)$. Finding this constant is related to the Kronecker limit formula for the derivative at $0$ of the zeta function $Z(s) = \sum \nolimits _{m,n}’{|m + n\tau {|^{ - 2s}}}$. We consider instead the zeta function without absolute values.