I have some issues when dealing with people working outside number theory, to motivate and justify in some sense the problems I am interested in. Mainly, here are the issues I do not know enough motivations about:
- why do we care about the error term in PNT? (beyond "we have an equivalent, it is therefore natural to know to what extent it is true")
- is there any application to these error terms inside number theory? (for instance finding gaps between primes or any result that would be better when improving the error term)
- is there any argument to say that we cannot beat Riemann hypothesis' square root savings?
Thanks in advance for any insight!