Hi,
referring to the RiemannSiegel approximate functional equation for Riemann's Zeta
$ \zeta(s) = \sum_{n\leq x}\frac{1}{n^s} \ + \ \chi(s) \ \sum_{n\leq y}\frac{1}{n^{1s}} \ + \ O(x^{\sigma}+ \ t^{\frac{1}{2}\sigma}y^{\sigma  1}) $
would anybody know of a similar result applying to the Dirichlet Eta function ?
$ \eta(s) = \sum_{n=1}^\infty\frac{(1)^{n1}}{n^s} = 1\frac{1}{2^s}+\frac{1}{3^s}\frac{1}{4^s}+\ldots $
I am interested in expressing the Dirichlet Eta function in terms of its partial sums, as well as in terms of the partials sums of its critical line symmetrical "twin". So, I am looking for an expression of this kind
$ \eta(s) = !(s) \ \sum_{n\leq x}\frac{(1)^{n1}}{n^s} \ + \ ?(s) \ \sum_{n\leq y}\frac{(1)^{n1}}{n^{1s}} \ + \ O( ........) $
where !(s) and ?(s) are functions yet unknown to me, and I am not even sure whether such an approximate functional equation might actually exist. I will greatly appreciate suggestions from anybody familiar with the subject. Many Thanks.



First of all, if you want just a single sum up to T, then just like with the zeta function you have an approximation: $$\eta(s)\sim\sum_{n=1}^{T/2\pi}\frac{(1)^{n1}}{n^s}$$ It turns out additionally that the full approximate functional equation (aka RiemannSiegel formula) holds for really anything where you have a functional equation. You can see, for example, "The approximate functional equation for a class of zetafunctions" by Chandrasekharan and Narasimhan. In this case, you can write the functional equation as relating $12^{s}+3^{s}\cdots$ and $0.5^{s}1.5^{s}+2.5^{s}\cdots$ (this form does not have an annoying $12^s$ term in it) so this latter term is what you find in the final formula: $$\eta(s) \sim \sum_{n\le x}\frac{(1)^{n1}}{n^s}  \chi(s)\sum_{n\le y}\frac1{(n1/2)^s}$$ Also, any of the methods for proving the formula for $\zeta$ carry over naturally to $\eta$ (sometimes more naturally), so let me know if you had a particular one in mind. 


Hi, the related question being: $ 2^{s} \chi(s) \ ( \ \frac{1}{2} \sum_{1}^{2N}\frac{1}{n^{1s}} \  \ \sum_{1}^{N}\frac{1}{n^{1s}} \ ) \  \ \frac{1}{\chi(s)} \sum_{N+1}^{2N}\frac{1}{n^s} \ \sim \ \eta(1s) \; \; \; \; ? $ 

