Analytic continuation of a weight-k real-analytic Eisenstein series

The primitive weight-k real-analytic Eisenstein series must be distinguished from the sum over all nonzero integer pairs. For positive even $k$, write $y=\mathrm{Im}\tau$ and definein the half-plane of absolute convergence. The equality follows by separating a nonzero pair into a positive integer times a primitive pair, with a factor two for simultaneous sign.

For fixed $\tau$, the all-pairs sum $G_{k,s}$ continues to an entire function of $s$. One way to see this is to introduce the weighted theta functionThe zero term vanishes. The weight is a harmonic polynomial, so its Gaussian Fourier transform has zero constant mode as well. Poisson summation then gives rapid decay at both endpoints, up to powers of $t$, making its Mellin transform entire. Termwise integration in the initial half-plane yieldsMultiplication by the entire reciprocal gamma function gives the claimed continuation. The primitive series instead continues meromorphically as $E_{k,s}=G_{k,s}/(2\zeta (2s+k))$; it must not inherit an unqualified entire-function claim after division by the Riemann zeta function.