The Riemann hypothesis as a sequence of surface to volume ratios

Abstract

In 1955 Beurling showed that the Riemann hypothesis is equivalent to a closure property of a certain class of functions in L 2[0,1]. In 1975, Ryavec obtained explicit zero free regions for ζ( s) in the critical strip; those regions depend on the norms of certain linear functionals in such a way that if those norms are unbounded, the Riemann hypothesis holds. In fact, as Bombieri later remarked, the converse is also true. As this paper demonstrates, the set, W, of these linear functionals can be written naturally as the countable nested union of certain subsets, W n , and the extremal norms on these subsets form a strictly increasing sequence, one which is unbounded if and only if the Riemann hypothesis is true. We further show that the nth extremal norm can be reinterpreted as a certain surface to volume ratio in n dimensions. The form of these ratios suggests that the nth extremal norm is on the order of n —a conjecture reinforced by an extensive computer study. This study almost certainly reveals the extrema for cases n=2,3,4,5, although these are still unsolved problems.