Integrated correlation functions in \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory with gauge group SU(N) can be expressed in terms of the localised S4 partition function, ZN, deformed by a mass m. Two such cases are \( {\mathcal{C}}_N={\left(\operatorname{Im}\tau \right)}^2{\partial}_{\tau }{\partial}_{\overline{\tau}}{\partial}_m^2\log {\left.{Z}_N\right|}_{m=0} \) and \( {\mathcal{H}}_N={\partial}_m^4\log {\left.{Z}_N\right|}_{m=0} \) , which are modular invariant functions of the complex coupling τ. While \( {\mathcal{C}}_N \) was recently written in terms of a two-dimensional lattice sum for any N and τ, \( {\mathcal{H}}_N \) has only been evaluated up to order 1/N3 in a large-N expansion in terms of modular invariant functions with no known lattice sum realisation. Here we develop methods for evaluating \( {\mathcal{H}}_N \) to any desired order in 1/N and finite τ. We use this new data to constrain higher loop corrections to the stress tensor correlator, and give evidence for several intriguing relations between \( {\mathcal{H}}_N \) and \( {\mathcal{C}}_N \) to all orders in 1/N. We also give evidence that the coefficients of the 1/N expansion of \( {\mathcal{H}}_N \) can be written as lattice sums to all orders. Lastly, these large N and finite τ results are used to accurately estimate the integrated correlators at finite N and finite τ.
