I'm [crossposting](https://math.stackexchange.com/questions/3759310/functional-equation-for-etas-following-riemanns-2nd-method).


Being
\begin{equation\*}
\eta(s)=\sum\_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^{s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots
\end{equation\*}
and following Riemann's second method, (Edwards p.15), to obtain the functional equation for $\zeta(s)$ one
can think the same way and try the same aproach for $\eta(s)$.
Thus from
\begin{equation\*}
\int\_{0}^{\infty} \operatorname{exp}\left(-n^{2} \pi x\right) x^{s / 2-1} d x=\pi^{-s / 2} \Gamma\left(\frac{s}{2}\right)\frac{1}{n^{s}} \text { for } s>0
\end{equation\*}
one can express $\eta(s)$ as
\begin{equation\*}
\pi^{-s / 2} \Gamma \left(\frac{s}{2}\right)\underbrace{\left(1-\frac{1}{2^{s}}+\frac{1}{3^{s}}+\cdots\right)}\_{\eta(s)} =\int\_{0}^{\infty}\left(e^{-\pi 1^2 x}-e^{-\pi 2^2 x}+e^{-\pi 3^2 x}+\cdots\right)x^{s/2}\text{ }\frac{dx}{x}
\end{equation\*}
How would one proceed from here to craft a functional equation for $\eta(s)$?


I'm interested in refferences and/or answers. Any of them will be very much apreciated.


Thanks.



Functional equation for $\eta(s)$ following Riemann's $2^{nd}$ method
Ignoring technicalities of convergence, in Riemann's second proof, you start with the Poisson summation formula $\sum\_{n\in\mathbb Z} f(n / x) = x \sum\_{n\in\mathbb Z} \hat f (n x)$, take the Mellin transform of both sides, and use the self-dual function $f(x)=e^{-x^2}$.


To get the alternating sum you want, you could either change the function or change the summation formula. For the function, you could use something like $\sum\_{n\in\mathbb Z} f(n / x) \exp(\pi i n)$, and do some computations. You could also take a twisted Poisson summation formula $\sum (-1)^n f(n) = \sum\_{n \textrm{ odd}} \hat f(n/2)$, but the steps for proving that are identical to the manipulations done to derive the functional equation for $\eta(s)$ from the functional equation for $\zeta(s)$.


Furthermore, an inverse Mellin transform allows you to go in the converse direction: a functional equation of Dirichlet series gives a summation formula. If the gamma factor is different then it will not be a Fourier transform but a generalization. If the degree of the functional equation is $d$ then the sum will be over $d$-th roots of natural numbers instead of over natural numbers.


