 Kronecker limit formula

This normalization of the first Kronecker limit formula follows from the Fourier expansion of the completed nonholomorphic Eisenstein series. At $s=1$, $K_{1/2}(t)=\sqrt{\pi /(2t)}{e}^{-t}$, so the nonconstant terms sum to $-2\log |{\prod }_{m\ge 1}(1-q^m)|$. The constant terms supply $\pi y/6-(\log y)/2+(\gamma -\log (4\pi ))/2$. Combining them gives the Dedekind eta function expression, with $\gamma$ the Euler--Mascheroni constant.

The Kronecker limit formula is an important result in the theory of modular forms and number theory. It relates the behavior of certain L-functions to the special values of those functions at integers. Specifically, it provides a way to compute the special value of an L-function associated with a point on a certain modular curve. The formula can be stated in the context of the Dedekind eta function and the Eisenstein series.