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We quantify the set of known exponent pairs $(k, \ell)$ and develop aframework to compute the optimal exponent pair for an arbitrary objectivefunction. 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 $\zeta(1/2 + it)$, zero-density estimates of$\zeta(s)$ and the generalised Dirichlet divisor problem.

On optimal exponent pairs