Nonholomorphic Eisenstein series ($G(\tau ,s)$)

For $\mathrm{Re}s>1$,is an absolutely convergent modular-invariant function.

This modular-invariant function, initially defined for $\Re s>1$, has Fourier expansionwhere $\Xi (u)={\pi }^{-u/2}\Gamma (u/2)\zeta (u)$ is the completed Riemann zeta function, without the entire-xi polynomial factor. Poisson summation and the integral representation of the modified Bessel function of the second kind prove the expansion. It gives meromorphic continuation, $\mathcal{E}(z,s)=\mathcal{E}(z,1-s)$, and simple poles at zero and one with residues $-1/2,1/2$. Apparent poles at $s=1/2$ cancel between the two constant terms.

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.

For $\mathrm{Re}s>1$ this Eisenstein series also equals ${\sum }_{\gamma \in {\Gamma }_{\infty }\backslash S{L}_2(\mathbb{Z})}(\mathrm{Im}\gamma \tau {)}^s$, where ${\Gamma }_{\infty }$ consists of translations and their negatives. It is modular invariant. The nonholomorphic Eisenstein series summed over all nonzero integer pairs instead equals $2\zeta (2s)E(\tau ,s)$: separate each pair into a positive integer times a primitive pair. This normalization matters in Rankin–Selberg unfolding.

For the series over all nonzero integer pairs, the constant Fourier coefficient is $2\zeta (2s)y^s+2\sqrt{\pi }\Gamma (s-1/2)\zeta (2s-1)y^{1-s}/\Gamma (s)$. Integrating over one real period unfolds the translated intervals and then uses the Gaussian integral. Multiplication by ${\pi }^{-s}\Gamma (s)$ expresses it as $2\Lambda (2s)y^s+2\Lambda (2s-1)y^{1-s}$ in terms of the completed Riemann zeta function.

For the normalization $(T_{p}f)(\tau )=p^{-1}(f(p\tau )+{\sum }_{b\mathrm{mod}p}f((\tau +b)/p))$, one has