Eh, someone just suggested that a bound of this form might break some cryptographic assumptions for finite fields of (large) prime order. (This is not obvious t

Eh, someone just suggested that a bound of this form might break some cryptographic assumptions for finite fields of (large) prime order. (This is not obvious to me, but it's a point that I think is worth considering, and already would show that such a result is useful.) It's very hard to know how these things end up being useful. Even if a specific piece of knowledge isn't useful by itself, it can lead to things that are useful down the line. It's both very hard to know which ones will be, or which ones do lead to actually useful results; for example, think of work on propositional logic which ends up being the bedrock of computation, work on elliptic curves (cryptography), work on PDEs (physics), quantum physics (transistors), photonics (communications), etc. Some fields have pretty "fast-to-application" times, but it's not always clear which ones will _not_ yield useful results. Plus, and here's the real, honest answer: all of this shit is very fun! Why not do it? I'm also a (somewhat) practical person, and that guides many of the problems I try to solve, but sometimes there's just such a tantalizing puzzle that it's hard not to resist! It's pretty damn hard to "only" work on "useful" stuff and not be just an ok academic/programmer/etc., you have to let yourself get nerd-sniped into doing random things. With very high probability, these little "side-quests" end up being very useful down the line, for one reason or another.