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Concerning Hardy-Littlewood approximate functional equation for the [tex] \zeta [/tex] function
[tex] \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}) [/tex]
does somebody know of any similar result for the Dirichlet [tex] \eta [/tex] function ? where [tex] \eta (s) [/tex] is defined as
[tex] \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 [/tex]
[tex] \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}) [/tex]
does somebody know of any similar result for the Dirichlet [tex] \eta [/tex] function ? where [tex] \eta (s) [/tex] is defined as
[tex] \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 [/tex]