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?
Best Answer
A numerical counterexample to the first conjecture is $$ t = 282.4547208234621746108397940690599354\ldots $$ where both gp and Wolfram Alpha 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 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).