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)