The over ~1½ century old Riemann Hypothesis has deep implications in mathematics and a large edifice of math theory is now proved conditionally on it and numerous variants. I recently came across a reference to a conditional result in TCS based on the Riemann hypothesis. I am therefore wondering,

what are the major implications of the Riemann hypothesis in TCS?

As a start here is an example from a recent paper, Homomorphism Polynomials complete for VP by Durand, Mahajan, Malod, de Rugy-Altherre, and Saurab. From the paper's introduction:

One of the most important open questions in algebraic complexity theory is to decide whether the classes VP and VNP are distinct. These classes, first defined by Valiant in [13, 12], are the algebraic analogues of the Boolean complexity classes P and NP, and separating them is essential for separating P from NP (at least non-uniformly and assuming the generalised Riemann Hypothesis, over the field $\mathbb{C}$, [3]).