Finite sum seemingly related to nontrivial zeta zeros

Question

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

$t=10^6$.

Answer 1

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.

Answer 2

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.