Playing with the Riemann zeta function I came across a property of partial sums of $\zeta(s)$ that I wonder if anyone could explain. Taking a partial sum as$$\zeta_{ab}(s)=\sum_{n=a}^b\frac{1}{n^s}$$
where $s={1\over2}+it$
If $a=\lfloor{{t}\over{(2k+1)\pi}}\rfloor$ and $b=\lfloor {{t}\over{(2k-1)\pi}}\rfloor$ then
$$|\zeta_{ab}(s)|\approx{1\over{\sqrt{k}}}$$ For large $t$ anyway. This relates to the dimensions of the spirals formed by the partial sums.
Is there a way to derive that?
As an example, here's the progression of $|\zeta_{ab}(s)|$ for $t=10000$ and $k=1$:
...and $k=2$:
I also investigated this property of partial sums of the Dirichlet series and came to the conclusion that it corresponds to the functional equation of the Riemann Zeta function. The vectors corresponding to the members of the Dirichlet series form a polyline, which I called the Riemann spiral, then the segments connecting the centers of the Riemann spiral (I called their meddle vectors) form another polyline, which I called the inverse Riemann spiral. Then we can write the following equation (taking into account the generalized summation):
$\sum_{n=1}^\infty\frac{1}{n^s}-(\frac{t}{2\pi})^{1/2-\sigma}e^{-i\alpha_1}\sum_{n=1}^\infty\frac{1}{n^{1-s}}=0$
where $\alpha_1=t\log(\frac{t}{2\pi})-t-\frac{\pi}{4}+O(t^{-1})$ - the angle of the first middle vector
inverse Riemann spiral
I don't have enough reputation to answer, but I have the results of the study arXiv:1910.08363