Hi, referring to the Riemann-Siegel 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^{1-s}} \ + \ 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)^{n-1}}{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)^{n-1}}{n^s} \ + \ ?(s) \ \sum_{n\leq y}\frac{(-1)^{n-1}}{n^{1-s}} \ + \ 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.

Hi, referring to the Riemann-Siegel 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^{1-s}} \ + \ 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)^{n-1}}{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)^{n-1}}{n^s} \ + \ ?(s) \ \sum_{n\leq y}\frac{(-1)^{n-1}}{n^{1-s}} \ + \ 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.