Talk:Dirichlet Series Convergence Lemma/Lemma

The sublemma as stated cannot be true as it would mean an ordinary dirichlet series is uniformly convergent on any halfplane bounded away from its abscissa of convergence, which is not true. After checking the cited source, the correct version of the lemma is $2 M \left(1+ \dfrac {\left\vert s-s_0 \right\vert} {\sigma-\sigma_0} \right) n^{\sigma-\sigma_0}$

I should note that if desired, I can modify the last line of the new proof of the main lemma to give the result $M \dfrac {\left\vert s-s_0 \right\vert} {\sigma-\sigma_0} n^{\sigma-\sigma_0}$ which is slightly stronger. Upon inspecting the proof given in the cited source, it is essentially equivalent to the one I recently wrote (the difference being that I used Abel's lemma, while they used Abel's summation formula).

There are two possible ways to fix the proof of this sublemma, both of which put into question the purpose of doing so.

I could just copy the proof of the lemma as stated in the cited source; however, I'd be writing a proof functionally equivalent to the one I gave that is only applicable to ordinary Dirichlet series and not even as strong of a result as the new proof gives.

If instead I use the new proof to justify this sublemma, and in fact improve it slightly, so that it can then be used to prove the main lemma for ordinary Dirichlet series, it begs the question of why we don't just use that the proof given for the general case implies the result for ordinary Series from the outset.

What do you guys think is the best course of action? Personally, Isn't generalizing's main use to prove a theorem for a lot different constructs at once? For example, the new proof implies the existence of the radius of converence of a power series aswell as the halfplane of convergence for ordinary dirichlet series. -- AliceInNumberland (talk) 22:53, 27 May 2018 (EDT)

$\mathsf{Pr} \infty \mathsf{fWiki}$ supports multiple proofs of any given result. Hence it is appropriate to have two proofs for the convergence of ordinary dirichlet series: one which applies the result for the general case, and one which uses the non-general technique.

A problem with abstraction is that it can take a result out of a realm of elementary mathematics into one requiring the mastery of more advanced concepts. Hence a student at a lower level (high school or undergraduate) who is well placed to comprehend a proof based in, say, the domain of real numbers, may be out of their depth to encounter a proof of such a result which requires the knowledge of abstract algebra, linear algebra, and so on. We aim to allow for more elementary approaches to exist side by side with those of a more advanced nature which fully generalise the initial concept. A case in point is that of polynomials. A basic approach where we initially defined a polynomial as a finite power series in the real numbers got abstracted to a general algebraic construct in multiple indeterminates over an arbitrary field (or even ring, I misremember), which left, for example, the elementary continuity proofs in real analysis high and dry.

As for perceived errors: if a page is genuinely incorrect (as this one may indeed be -- I have not taken the time to study it as analysis bores me), then our approach is not to delete it, but to correct it. This is one of the reasons we have for providing references to source works, so that any material on this site (as is the case here) can be cross-checked against those source works. Works which have been published in hard-copy are greatly preferred to online resources, as the latter can prove ephemeral (Khan Academy is a case in point: a restructure of that website has meant that all our links to it are now broken). --prime mover (talk) 04:34, 28 May 2018 (EDT)