Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 2 / d / Solution

Set $r=k-l$. Since $k$ and $l$ are even and $k\ge l+6$, one has $r\ge 6$. The holomorphic Eisenstein series in the question is absolutely convergent and decomposes asbecause every nonzero integer pair is a positive multiple of a primitive pair and the two signs contribute the factor two.

Absolute convergence, including that established in part (c) at $s=k-1$, permits Rankin–Selberg unfolding. Unfolding the Petersson inner product from the fundamental domain to the strip $0\le x<1$, $y>0$, givesThe $x$-integral uses orthogonality of complex exponentials to retain equal Fourier indices:Finally,Substitution gives the Rankin–Selberg unfolding identity for a holomorphic Eisenstein series