I am writing my master degree work, about numerical approximation for the Riemann Zeta function. In my work I created an approximation that is written below but I believe, that it's identity.
Where:
$z = a + xi$
$\displaystyle \zeta(z) = \frac{1}{1-2^{1-z}}\eta(z)$
$E_n(x)$ - generalized exponential integral
Then:
$$\begin{align} \eta(z) &\approx \frac{1}{2} - \sum_{n=1}^\infty (E_z(-i \pi(2n-1))+E_z(i\pi(2n-1))) \\&= \frac{1}{2} - 2 \sum_{n=1}^\infty \int_1^\infty {\frac{\cos(\pi (2n-1)t)}{t^z}} \, dt \end{align}$$
Why I believe that it is identity, not approximation ?
This approximation is based on researches about partial sum of Dirichlet Eta function. On the photo that is linked below, you can see that I found a function that represents all separated steps and also I know on which values this steps shows. These examples are actually for $z=\frac{1}{2}+200\pi$i. The Red line is the real part, and the blue line is the imaginary part of partial sum of Eta function. My functions are colored in cyan and magenta.
I calculated this integral by mathematica and it doesn't look pretty well. I believe that there should be a better way for it.
Question (1): Do these functions remind anyone of any mathematical or physical formulas, etc. ?
Question (2): If this approximation will be globally convergent to Zeta function, how to show that symbolically if it's even possible ?