  In the workshop of the July 2016 Building Bridges 3 conference in Sarajevo, I
presented the results from a joint article with W. Raji (Mathematische Annalen
2017, preprint arXiv:1402.1854). That article gave a proof of various linear
relations between products of two Eisenstein series on $\Gamma(N)$, including
an interesting identity related to the action of a Hecke operator on such a
product. The real proofs involve some care to deal with issues of convergence.
In this note we give "fake" proofs for these identities, ignoring the
convergence issues; some of these fake proofs appeared in the workshop lecture
as an amusing side note before I sketched the real proofs. Something in these
fake proofs is quite suggestive, even though the proofs themselves are clearly
invalid (and even produce wrong results). It would be interesting to understand
what exactly is going on here.
