Finite sum seemingly related to nontrivial zeta zeros For $t \in \mathbb{R}$ define
$$ F(t) = \sum_{n=1}^{[t]} \frac{(-1)^{(n-1)}}{n^{\frac12 + it}}$$
Let $\operatorname{Arg}(t)$ be $\operatorname{atan2}(\Im t , \Re t)$ -
basically this is $\arctan$, but the sign depends on the quadrant.
Observation $\operatorname{Arg}(F(t))$ jumps (usually from negative
to positive) very near all nontrivial zeta zeros on the critical line
and in seemingly rare occasions without zeros. Computing $F(t)$ the
naiive way is not efficient for me.
$F(t)$ is truncated Dirichlet eta function on the critical line,
but it is not $0$ at zeros of zeta, though $|F(t)|$ has local minima near zeros.
Added Wolfram Alpha found closed form
for $F(t)$ in terms of Hurwitz zeta and zeta:
\begin{align} 
& \sum_{n=1}^k\frac{(-1)^{n-1}}{n^{\frac12 + i t}} = \\ 
& 2^{-1/2-i t}(-(-1)^k \zeta(i t+1/2, (k+1)/2)+ \\
& (-1)^k \zeta(i t+1/2, (k+2)/2)+2^{1/2+i t} \zeta(1/2 i (2  t-i))-2 \zeta(1/2 i (2 t-i)))
\end{align}
Setting $k=[t]$ gives $F(t)$. Numerical evidence supports the closed form.
In comments Greg Martin suggested $F(t)$ might not be correlated to
higher zeros, though numerical evidence suggests it is correlated
at height $10^6$, including closely spaced zeros.
Another observation is $|F(t)|$ appears to have local minima
close to zeta zeros on the critical line.
Setting $$ G(t) = \sum_{n=1}^{[t]} \frac{(-1)^{(n-1)}}{n^{1 + it}}$$
the jumps of $G(t)$ appear zeros of $\eta(1+i t)$ and looks like
$|G(t)| \sim |\eta(1 + i t)|$

Can this be explained?
Counterexamples?

Plot:

Closely spaced zeros and modulus

$t=10^6$.

 A: I'm not sure why this is mysterious.  The function $F(t)$ is a partial sum for the "$\eta$-function"
$$
\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^{1/2+i t}}=(1-2^{1-s})\zeta(s)
$$
a function which converges conditionally, and uniformly for $t$ in a compact set.  Now $F(t)$ has discontinuities due to the truncation $[t]$.  However, for large $t$ where the density of zeros of $\eta$ is $2\pi/\log(t)$ we will see many zeros in between each discontinuity.
The function $\eta(1/2+it)$ will trace out a curve in the complex plane, passing repeatedly through the origin.  It's argument will have a discontinuity of $+\pi$ each time (I'm assuming RH here.)  Since $F(t)$ tracks this function closely, it's argument will change rapidly, even when $F(t)$ does not pass through the origin.
The places where $\arg F(t)$ seems to jump away from a Riemann zero are more mysterious, but I would first consider the possibility the $\arg$ code is lying to you.
A: About the local minima approximating zeros, have you considered Hurwitz theorem? if the limit of the sequence of partial sums is 0 (not identically), then, in a disk close neighbourhood of the concerned zero, each nth partial sum with n greater than an index k (k depending on the size of said disk) would feature a zero. Perhaps, your minima correspond indeed to said zeroes ... of course, in general, to exactly hit zero, instead of a minimum just close to it, you will have to move within said disk also a bit away from the critical line.   
