Toward optimal exponent pairs

Abstract

We quantify the set of known exponent pairs ( k , ℓ ) (k, \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 ζ ( s ) \zeta (s) in the critical strip, estimates of the moments of ζ ( 1 / 2 + i t ) \zeta (1/2 + it) and the generalised Dirichlet divisor problem.