How to improve the quality and visibility of research?

Red makes your glove shoot out fireballs, whereas pink heals you.


Amazingly broad question but a couple of simple answers.

  • Indexing like you suggested. Make sure the appropriate indexes exist and if the need is great enough make sure they are all covering.

  • Get rid of the *. Specify the columns you need. If you are pulling across 100 columns over a million rows that's going to be a LOT of data. If you only need 3 columns only specify 3 columns.

  • Change the timeout. This sounds hokey but honestly sometimes it really is the answer. SQL Server (for example) doesn't actually have a timeout so the problem is on the connection side. Have them increase the timeout to 30 seconds (assuming this is an acceptable amount of time, and it frequently won't be).

  • Make sure it's the DBMS's fault. It's very possible you are having a problem on the connection side that's causing it to take 5-10 seconds just to connect. Fix that and you are well within your time.


The result evidently follows only from absolute convergence of $Q(s)$. As I mentioned in the question, any product of two absolutely convergent Dirichlet series is an absolutely convergent Dirichlet series, so let

$$ \frac{Q^k(s)}{k!} = \sum_{n=1}^{\infty} \frac{a_{k,n}}{n^s}. $$

Then

$$ e^{Q(s)} = \sum_{k=0}^{\infty} \frac{Q^k(s)}{k!} = \sum_{k=0}^{\infty} \sum_{n=1}^{\infty} \frac{a_{k,n}}{n^s}. $$

If we assume $Q(s)$ converges absolutely, then the inner sum converges asbolutely (after which the outer sum clearly does too), so we may swap the order of summation to get

$$ e^{Q(s)} = \sum_{n=1}^{\infty} \sum_{k=0}^{\infty} \frac{a_{k,n}}{n^s} = \sum_{n=1}^{\infty} \frac{b_n}{n^s}, $$

where

$$ b_n = \sum_{k=0}^{\infty} a_{k,n}. $$

Conversely, if $\sum_n \frac{b_n}{n^s}$ converges absolutely, then, since the $a_{k,n}$ are all nonnegative, the double sum $\sum_n \sum_k \frac{a_{k,n}}{n^s}$ converges absolutely, and hence switching the order of summation allows us to conclude that each $\sum_n \frac{a_{k,n}}{n^s}$ converges absolutely. In particular, $Q(s)$ converges absolutely, as desired.