TITLE: Heuristic argument for the Riemann Hypothesis
QUESTION [33 upvotes]: Is there a heuristic argument that supports the validity of the Riemann hypothesis or are we just relying on numerical evidence?  Moreover, what is the strongest theorem that supports the validity of RH?

REPLY [13 votes]: I am not sure that this is a proper answer for your question. Sorry in advance if it is not.
Other answers provided some good examples of heuristic arguments supporting the RH. But I want to point at a heuristic result that puts some serious doubts on it. Since I am not so familiar with the technical details, it is better to quote directly from this answer:

The De Bruijn-Newman constant $\Lambda$ was defined and upper bounded by $\Lambda\le\frac{1}2$ in 1950. After 58 years of work, in 2008 this upper bound was finally improved to... $\Lambda<\frac{1}2$ (a 0% improvement) in a 26-page paper. The best known upper bound is currently $\Lambda<0.22$. The Riemann hypothesis was known to be equivalent to $\Lambda\le 0$, so if it's true then we've got quite a ways to go.

But Terrence Tao and Brad Rogers recently proved that $\Lambda\ge 0$. So in Tao's words (actually, Newman's words quoted by Tao):

If the Riemann hypothesis is true, then it's 'just barely' true.