Modified commutation relationships via the Riemann Hypothesis

Current approaches to quantum gravity suggest there should be a modification of the standard quantum mechanical commutator, $[{\hat x} , {\hat p}] = i \hbar$. Typical modifications are phenomenological and designed to result in a minimal length scale. As a motivating principle for the modification of the operators and their commutators, we assume the validity of a version of the Bender-Brody-M\"uller variant of the Berry-Keating approach to the Riemann hypothesis. We arrive at a family of modifications of the position and momentum operators which lead to a minimal length scale $\Delta x_0$. Additionally, this larger family satisfies and generalizes the structure of the Bender-Brody-M\"uller approach to the Riemann hypothesis.