There is a vast array of mathematical results that hold _if_ Riemann is true. Wikipedia has a list: https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequenc

There is a vast array of mathematical results that hold _if_ Riemann is true. Wikipedia has a list: https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequences https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequence... Essentially, it seems to encode a hard ground truth on the behaviour (and in particular the growth) of arithmetic structures. It often (but not always) is the case that Riemann gives you a best/strongest version of a result. Meaning that the zeta function in some meaningful way encodes all there is to know.