TITLE: Rational Roots of Riemann's $\zeta$ Function
QUESTION [12 upvotes]: A look at the first few zeros 
$$14.134725,21.022040,25.010858,30.424876,32.935062,37.586178,\dots$$
is in accord with

Numerical evidence suggests that all values of $t$ (the imaginary part of a root of $\zeta$) corresponding to nontrivial zeros are irrational (e.g., Havil 2003, p. 195; Derbyshire 2004, p. 384). 

(numbers and quote taken from here). What are the attempts to prove that all values of $t$ are irrational? Would it mean something to the distribution of primes, if one, some or plenty of rational roots $\frac{1}{2}+i\frac{q}{r}$ exist?

REPLY [14 votes]: Actually, the rationality or irrationality of the Riemann zeros does have subtle influence on the distribution of primes.  In analytic number theory, this sub-subject goes under the name 'Oscillation Theorems'.  An example can be found in the (excellent) book "Multiplicative Number Theory" by Montgomery and Vaughan.  Corollary 15.7 says that if the ordinates $\gamma>0$ of the Riemann zeros are linearly independent over $\mathbb Q$, then
$$
\limsup_{x\to\infty}\frac{M(x)}{x^{1/2}}=+\infty
$$
and
$$
\liminf_{x\to\infty}\frac{M(x)}{x^{1/2}}=-\infty.
$$
Here $M(x)$ is the summatory function of the Möbius $\mu$ function:
$$
M(x)=\sum_{n<x}\mu(n).
$$
The connection is, of course, the explicit formula.
Edit:  As a second example, Rubinstein and Sarnak show (roughly speaking) that under the Riemann Hypotheis and the assumption the zeros are linearly independent over $\mathbb Q$, that
$$
\lim_{x\to\infty}\frac{1}{\log(x)}\sum_{\substack{n<x\\\pi(n)\ge \text{Li}(n)}}\frac{1}{n}=0.00000026
$$
as well as other results, in their paper Chebyshev's Bias.