Modified commutation relationships from the Berry-Keating program

Current approaches to quantum gravity suggest there should be a modification of the standard quantum mechanical commutator, $[\widehat{x},\widehat{p}]=i\hslash$. Typical modifications are phenomenological and designed to result in a minimal length scale. As a motivating principle for the modification of the position and momentum commutator, we assume the validity of a version of the Bender-Brody-Müller variant of the Berry-Keating approach to the Riemann hypothesis. We arrive at a family of modified position and momentum operators, and their associated modified commutator, which lead to a minimal length scale. Additionally, this larger family generalizes the Bender-Brody-Müller approach to the Riemann hypothesis.