The $D^6 R^4$ interaction as a Poincar\'e series, and a related shifted convolution sum

We complete the program, initiated in a 2015 paper of Green, Miller, and Vanhove, of directly constructing the automorphic solution to the string theory $D^6 R^4$ differential equation $(\Delta-12)f=-E_{3/2}^2$ for $SL(2,\Z)$. The construction is via a type of Poincar\'e series, and requires explicitly evaluating a particular double integral. We also show how to use double Dirichlet series to formally derive the predicted vanishing of one type of term appearing in $f$'s Fourier expansion, confirming a conjecture made by Chester, Green, Pufu, Wang, and Wen motivated by Yang-Mills theory (and later proved rigorously by Fedosova, Klinger-Logan, and Radchenko using the Gross-Zagier Holomorphic Projection Lemma.).