jats:titleAjats:scbstract</jats:sc> </jats:title>jats:pWe present a novel expression for an integrated correlation function of four superconformal primaries in SU(jats:italicN</jats:italic>) jats:inline-formulajats:alternativesjats:tex-math$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miN</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> = 4 supersymmetric Yang-Mills (jats:inline-formulajats:alternativesjats:tex-math$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miN</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> = 4 SYM) theory. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. In this paper the correlator is re-expressed as a sum over a two dimensional lattice that is valid for all jats:italicN</jats:italic> and all values of the complex Yang-Mills coupling jats:inline-formulajats:alternativesjats:tex-math$$ \tau =\theta /2\pi +4\pi i/{g}{\mathrm{YM}}^2 $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miτ</mml:mi> mml:mo=</mml:mo> mml:miθ</mml:mi> mml:mo/</mml:mo> mml:mn2</mml:mn> mml:miπ</mml:mi> mml:mo+</mml:mo> mml:mn4</mml:mn> mml:miπi</mml:mi> mml:mo/</mml:mo> mml:msubsup mml:mig</mml:mi> mml:miYM</mml:mi> mml:mn2</mml:mn> </mml:msubsup> </mml:math></jats:alternatives></jats:inline-formula>. In this form it is manifestly invariant under SL(2jats:italic,</jats:italic> ℤ) Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the SU(jats:italicN</jats:italic>) correlator to the SU(jats:italicN</jats:italic> + 1) and SU(jats:italicN −</jats:italic> 1) correlators. For any fixed value of jats:italicN</jats:italic> the correlator can be expressed as an infinite series of non-holomorphic Eisenstein series, jats:inline-formulajats:alternativesjats:tex-math$$ E\left(s;\tau, \overline{\tau}\right) $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miE</mml:mi> mml:mfenced mml:mis</mml:mi> mml:miτ</mml:mi> mml:mover mml:miτ</mml:mi> mml:mo¯</mml:mo> </mml:mover> </mml:mfenced> </mml:math></jats:alternatives></jats:inline-formula> with jats:italics</jats:italic> ∈ ℤ, and rational coefficients that depend on the values of jats:italicN</jats:italic> and jats:italics</jats:italic>. The perturbative expansion of the integrated correlator is an asymptotic but Borel summable series, in which the jats:italicn</jats:italic>-loop coefficient of order (jats:italicg</jats:italic>jats:subYM</jats:sub>/jats:italicπ</jats:italic>)jats:sup2jats:italicn</jats:italic></jats:sup> is a rational multiple of jats:italicζ</jats:italic>(2jats:italicn</jats:italic> + 1). The jats:italicn</jats:italic> = 1 and jats:italicn</jats:italic> = 2 terms agree precisely with results determined directly by integrating the expressions in one-loop and two-loop perturbative jats:inline-formulajats:alternativesjats:tex-math$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miN</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> = 4 SYM field theory. Likewise, the charge-jats:italick</jats:italic> instanton contributions (|jats:italick</jats:italic>| = 1jats:italic,</jats:italic> 2jats:italic, . . .</jats:italic>) have an asymptotic, but Borel summable, series of perturbative corrections. The large-jats:italicN</jats:italic> expansion of the correlator with fixed jats:italicτ</jats:italic> is a series in powers of jats:inline-formulajats:alternativesjats:tex-math$$ {N}^{\frac{1}{2}-\mathrm{\ell}} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msup mml:miN</mml:mi> mml:mrow mml:mfrac mml:mn1</mml:mn> mml:mn2</mml:mn> </mml:mfrac> mml:mo−</mml:mo> mml:miℓ</mml:mi> </mml:mrow> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> (jats:italicℓ</jats:italic> ∈ ℤ) with coefficients that are rational sums of jats:inline-formulajats:alternativesjats:tex-math$$ E\left(s;\tau, \overline{\tau}\right) $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miE</mml:mi> mml:mfenced mml:mis</mml:mi> mml:miτ</mml:mi> mml:mover mml:miτ</mml:mi> mml:mo¯</mml:mo> </mml:mover> </mml:mfenced> </mml:math></jats:alternatives></jats:inline-formula> with jats:italics</jats:italic> ∈ ℤ + 1/2. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider the ’t Hooft topological expansion of large-jats:italicN</jats:italic> Yang-Mills theory in which jats:inline-formulajats:alternativesjats:tex-math$$ \lambda ={g}{\mathrm{YM}}^2N $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miλ</mml:mi> mml:mo=</mml:mo> mml:msubsup mml:mig</mml:mi> mml:miYM</mml:mi> mml:mn2</mml:mn> </mml:msubsup> mml:miN</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> is fixed. The coefficient of each order in the 1jats:italic/N</jats:italic> expansion can be expanded as a series of powers of jats:italicλ</jats:italic> that converges for jats:italic|λ| < π</jats:italic>jats:sup2</jats:sup>. For large jats:italicλ</jats:italic> this becomes an asymptotic series when expanded in powers of jats:inline-formulajats:alternativesjats:tex-math$$ 1/\sqrt{\lambda } $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mn1</mml:mn> mml:mo/</mml:mo> mml:msqrt mml:miλ</mml:mi> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula> with coefficients that are again rational multiples of odd zeta values, in agreement with earlier results and providing new ones. We demonstrate that the large-jats:italicλ</jats:italic> series is not Borel summable, and determine its resurgent non-perturbative completion, which is jats:inline-formulajats:alternativesjats:tex-math$$ O\left(\exp \left(-2\sqrt{\lambda}\right)\right) $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miO</mml:mi> mml:mfenced mml:mrow mml:moexp</mml:mo> mml:mfenced mml:mrow mml:mo−</mml:mo> mml:mn2</mml:mn> mml:msqrt mml:miλ</mml:mi> </mml:msqrt> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>