I'm studying infinite series, but I'm a physicist, not a mathematician.
I got it from Hardy's THE GENERAL THEORY OF
DIRICHLET'S SERIES that the Dirichlet series $\eta(s) = 1^{-s} – 2^{-s} + 3^{-s} – \dots$, with $s = \sigma + it$, converges for $\sigma > 0$, and converges absolutely for $\sigma > 1$, and so it converges only conditionally on the critical strip.
But, by Riemann's theorem, you can make a conditionally converging series converge to any value you like by finding some suitable rearrangement of terms.
So, how can we ever be sure of any value of $\eta(s)$ calculated inside the critical strip? For instance, if we calculate $\eta(s)$ at a non-trivial zero of the Riemann zeta function, $\zeta(s)$, it gives us zero for one arrangement of terms, but may give a different value for other arrangments. Likewise, we may get a zero value for $\eta(s)$ at values of $s$ that are not actual zeros of $\zeta(s)$.
Best Answer
The function $\eta(s)$ for ${\rm Re}(s) > 0$ is defined as an infinite series with terms ordered in the way you specified. That some theorem (namely the Riemann rearrangement theorem) says other rearrangements of the terms could give other values is irrelevant because those rearranged series are not defined as $\eta(s)$, and nobody cares about those other series for the purpose of working with $\eta(s)$. It's sort of like asking how we can possibly know what $\pi$ is because moving the digits around (like changing 3.1415926535... to 3.4151295653...) leads to other numbers. Basically, so what? Such a change is introducing a distraction that has no importance to the object of actual interest.
It is true that the value of a conditionally convergent series is sensitive to the order of the terms while absolutely convergent series are not, but it's not as if a series being absolutely convergent is all we need to get good estimates on the value of the series. For example, the partial sums of the series $\sum_{n \geq 1} 1/n^2$ converge absolutely but rather slowly (let's ignore the fact that we have a nice exact formula $\pi^2/6$). There are series acceleration methods that can improve the rate of convergence of a transformed version of many infinite series, both absolutely and conditionally convergent ones.
In the case of $\eta(s)$, nobody would ever use the definition of $\eta(s)$ as an infinite series to do numerical computations with it. Instead, express $\eta(s)$ in terms of the zeta-function: $\eta(s) = (1 - 2/2^s)\zeta(s)$. This is good because $\zeta(s)$ can be described by many kinds of formulas, including some rapidly convergent formulas involving integrals.
By the way, can you indicate why you are interested in $\eta(s)$? It the reason is because you are interested in $\zeta(s)$, then focus on $\zeta(s)$ itself. The "accident" that $\eta(s)$ can be expressed by a series formula valid for ${\rm Re}(s) > 0$ when $\zeta(s)$ can't would not be a good reason for regarding $\eta(s)$ as more important or more interesting than $\zeta(s)$.