Toward optimal exponent pairs

We quantify the set of known exponent pairs $(k,\mathrm{\ell })$ and develop a framework to compute the optimal exponent pair for an arbitrary objective function. Applying this methodology, we make progress on several open problems, including bounds of the Riemann zeta-function $\zeta (s)$ in the critical strip, estimates of the moments of $\zeta (1\mathrm{/}2+it)$ and the generalised Dirichlet divisor problem.