Zeroes of real and imaginary parts of $\zeta(1+it)$ separately (if any for $t>1$ say) Is anything known about existence and/or location of such zeroes ?
 A: Theorem 11.9 in Titchmarsh (with $\sigma_0=1$) tells us the values of $\log(\zeta(1+it))$, $t>1$ are everywhere dense in the plane.  Here $\log(\zeta(1+it))$ is defined by continuation along this line from $\sigma>1$.  From this one can see the imaginary part of $\log(\zeta(1+it))$ assumes all integer multiples of $\pi$, and so the imaginary part of $\zeta(1+it)$ is zero infinitely often, and similarly for the real part.
A: The solutions of $\Re\zeta(1+it)=0$ are scarce.  They limit small intervals where
$\Re\zeta(1+it)<0$. The probability in the sense of the limit of the
quotient of the measure of the
set $\{0<t<T: \Re\zeta(1+it)<0\}$ by $T$  is $d(1)=(3.80\pm0.01)\times 10^{-7}$.
R. Brent, J. van de Lune and I show it in https://arxiv.org/abs/1112.4910
We compute the first 50 such small intervals the first was situated around $t=682112.9$ and
have length $0.0529$ and the last of these 50 is around $t=8299958.2327$ of length 0.0432.
These are the first 100 zeros of $\Re\zeta(1+it)=0$.
The solutions of $\Im\zeta(1+it)=0$ are at least $cT$ with $c\ge \pi/\log 2$, because there
are real lines parallel to the real axis  that cut the line $\sigma=1$
in the x-ray of $\zeta(s)$ at distances $\pi/\log 2$, and each line gives a zero.
There are other real lines that cut the line $\sigma=1$, they are increasing in number with
$t$.
Apart from this an intuitive idea can be obtained from my paper on x-ray's of $\zeta(s)$
https://arxiv.org/abs/math/0309433
