Integrated correlation functions in N=4 supersymmetric
Yang--Mills theory with gauge group SU(N) can be expressed in terms of the
localised S4 partition function, ZNβ, deformed by a mass m. Two such
cases are CNβ=(ImΟ)2βΟββΟΛββm2βlogZNββ£m=0β and HNβ=βm4βlogZNββ£m=0β, which are modular invariant functions of the complex coupling
Ο. While CNβ was recently written in terms of a
two-dimensional lattice sum for any N and Ο, HNβ 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 HNβ to any desired order in 1/N and finite
Ο. We use this new data to constrain higher loop corrections to the stress
tensor correlator, and give evidence for several intriguing relations between
HNβ and CNβ to all orders in 1/N. We also give
evidence that the coefficients of the 1/N expansion of HNβ can be
written as lattice sums to all orders. Lastly, these large N and finite
Ο results are used to accurately estimate the integrated correlators at
finite N and finite Ο.Comment: 30 pages plus appendices, 8 figure