From parametric plots of $\zeta \left( \frac{1}{2} + it \right)$ it seems to be the case that:  

(1) except for $\zeta(\frac{1}{2})$ the Riemann zeta function does not attain any negative real value on the critical line.  

(2) the curve $(t, \zeta(1/2+it))$ is dense in the complex plane.


Are these statements known to be false, if not, is there any proof affirming them?



Negative values of Riemann zeta function on the critical line.
A numerical counterexample to the first conjecture is
$$
t = 282.4547208234621746108397940690599354\ldots
$$
where both **gp** and [Wolfram Alpha](https://www.wolframalpha.com/input?i=zeta%28282.4547208234621746108397940690599354+*+i+%2B+1%2F2%29) agree that $\zeta(\frac12 + it)$ has negative real part
$\simeq -0.02763$ and negligible imaginary part, so the actual zero of ${\rm Im}(\zeta(\frac12+it))$ near $t=295.5839\ldots$ yields a negative value of $\zeta(\frac12+it)$.


This was found by approximating $\zeta'(\frac12+it)$ at each of the first "few" zeros of $\zeta$ tabulated by Odlyzko in [http://www.dtc.umn.edu/~odlyzko/zeta\_tables/zeros1](http://www.dtc.umn.edu/%7Eodlyzko/zeta_tables/zeros1) and looking near the first zero (the 127th overall) at which $\zeta'$ has negative imaginary part. There are $22$ such zeros of the $649$ zeros whose imaginary part lies in $[0,1000]$; there's probably a counterexample near each of those, e.g. looking around the second such zero (#136) yields
$$
t = 295.583906974228176092587915204356841\ldots
$$
with $\zeta(\frac12+it) \simeq -0.0169004$.


**EDIT** 1) Henry Cohn (in a comment below) provides **gp** code that looks for solutions in an interval by dividing it into segments $(t\_0, t\_0 + 0.01)$, testing whether ${\rm Im}(\zeta(\frac12+it))$ changes sign between the endpoints, and if so whether the real part is negative at the crossing. Extending his computation to $0 \leq t \leq 1000$ finds the expected $22$ solutions; in particular $282.45472+$ seems to be the first.


2. Once one has calculated an answer one can ask Google for its previous appearances. Google recognizes $282.45472$ from J.Arias-de-Reyna's paper "X-Ray of Riemann zeta function" (<http://arxiv.org/abs/math/0309433>) where it appears (to within $10^{-5}$) as the first counterexample to "Gram's law" — see the plot on page 26 (thick and thin curves show where $\zeta(s)$ is real and imaginary respectively).


