Heuristic argument for the Riemann Hypothesis 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?
 A: The Riemann hypothesis is true, if primes are random in certain ways.
A: There are many theoretical results that support the (generalized) Riemann hypothesis:


*

*zero density estimates for the zeros of certain $L$-functions

*infinitely many zeros on the critical line of certain $L$-functions

*nonnegativity of the central value of certain $L$-functions

*subconvex bounds for certain $L$-functions

*strong lower and upper bounds for certain moments of certain $L$-functions

*the (proven) Riemann hypothesis for the $L$-functions of algebraic varieties over finite fields


and so on. Each of these items is a research topic on its own, with hundreds of articles. Use Google, Wikipedia, MathSciNet etc. to learn more about them.
A: The function field model came up in GH from MO's answer, and then also in user54038's. I just want to add some detail to explain how good of an analogy the function field model is. 
The Riemann zeta function does not stand alone, but instead is the simplest member of a series of wider classes of zeta and $L$-functions of deep relevance in number theory - among them Dedekind zeta funcitons, Dirichlet $L$-functions, Hecke $L$-functions, Artin $L$-functions, elliptic curve $L$-functions, automorphic $L$-functions. (More precisely each of these fall into two wide classes, those being the motivic $L$-functions and the automorphic $L$-functions, which are conjectured to more-or-less agree, so you could also say there is a single class of $L$-functions with many interesting sub-classes). For all of these, the Riemann hypothesis has been conjectured. 
There are direct analogues of all these in the function field setting, with the interrelationships among them basically the same. The Riemann hypothesis was proven for all of them - the first couple by Weil, the later ones by Deligne, and the automorphic ones by Drinfeld, L. Lafforgue, and V. Lafforgue. 
Furthermore properties known and conjectured of these zeta and $L$-functions, in particular those to do with the distribution of the zeroes, match up in straightforward ways. As far as I know, all known results about the distribution of zeroes of $\zeta$ and $L$-functions can be proven also over function fields by essentially the same arguments, and much stronger extensions of these results, fitting what we conjecture over $\mathbb Q$, are known in the "large $q$ aspect".
So the function field check on the Riemann hypothesis is not just a single analogy between two functions but a web of related analogies that all seem remarkably consistent with each other.
A: 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.

A: Besides all the theorems already mentioned I would like to add the Bombieri-Vinogradov theorem. This can be roughly thought of as an averaged version of the Generalised Riemann Hypothesis and thus gives strong support for GRH. Even more, in many proves it can be used to replace the Riemann Hypothesis and so make theorems unconditional.
A: There have been some good answers already given but I want to note another aspect, namely a heuristic involving the Möbius function. Let $\mu(n)$ be the Möbius function. The Riemann Hypothesis is equivalent to the claim that for any $\epsilon >0$ one has that $$\sum_{1 \leq n \leq x}\mu(n) = O(x^{1/2+\epsilon}).$$ This equivalence stems from making an explicit integral for $1/\zeta(s)$ in terms of the Möbius function which converges up the the 1/2 line if one has the above bound on the sum. 
Now, it is reasonable to guess/hope/assume that $\mu(n)$ is essentially random in the sense that the non-zero values behave essentially like a fair coin with heads corresponding to 1 and tails corresponding to –1. It turns out that if one has a fair coin, and one keeps flipping it, then the difference between the number of heads and tails after $x$ flips will with probability 1 be $O(x^{1/2+\epsilon})$. So if the Möbius function behaves like a random coin flip, or even close to a random coin flip we should expect RH to hold with probability 1.
Note that this isn't the only thing satisfying about this framing of the Riemann Hypothesis. This sort of shows one major reason that RH is important: a lot of sieves and other ways to get a handle on the primes involve inclusion/exclusion arguments, which is essentially what $\mu(n)$ is for. So in a moral sense RH says that if one is doing inclusion/exclusion with primes, one cannot get very large deviations in how much at any given stage one needs to include or exclude.
(Edit to add: actually this is essentially the same sort of approach as noted in the link by Pace above.)  
A: Bombieri reviews the evidence for the Riemann hypothesis in his problem description for the Millenium prize. In section III, he gives the numerical evidence as one of three pieces of evidence, the second being results on the density in the complex plane of hypothetical counterexamples, and the third that "it is known that more than 40% of nontrivial zeros of $\zeta(s)$ are simple and satisfy the Riemann hypothesis". However, in section IV, he says that "It may be said that the best evidence in favor of the Riemann hypothesis derives from the corresponding theory which has been developed in the context of algebraic varieties over finite fields." He gives special attention to Deligne's theorem. So, the strongest evidence is not the numerical evidence or even a heuristic argument, but rather an analogous theorem about a different system. At least, that's Bombieri's opinion.
