Rational Roots of Riemann's $\zeta$ Function

Question

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?

Answer 1

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.

Answer 2

It is very easy to make conjectures that some numbers are irrational (as a general principle, it's a good bet that something is irrational unless there is a good reason for it to be rational), but with a few exceptions it's very hard to prove them. AFAIK there is no reason for any of the nontrivial zeros to have rational $t$, but no reasonable hope of proving any of these $t$ to be irrational.

Answer 3

To the best of my knowledge, rationality of the imaginary part of one/some/all of the non-trivial zeros of zeta would have no consequences for the distribution of primes. On the other hand, irrationality of the real part of even one of these zeros would be another thing entirely!

Answer 4

A.M. Odlyzko on the nontrivial zeros of the zeta function:

"We will...write $$\rho_{n}=1/2+i\gamma_{n}$$ ...Nothing is known about the $\gamma_{n}$, but they are thought likely to be transcendental numbers, algebraically independent of any reasonable numbers that have ever been considered."

Answer 5

There were some recent attempts to see what can be done with the current technology, see

http://arxiv.org/abs/1109.1788

http://arxiv.org/abs/1208.2684

But as you can tell, we are still very far from being able to prove that any zero of $\zeta(s)$ is irrational.

If you had finitely many exceptions to the Riemann Hypothesis and all of them had rational imaginary parts this would have very surprising consequences for the distribution of the primes (from the explicit formula, there would be some very strong periodicity phenomena).