Why such an interest for the error term in the Prime Number Theorem 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!
 A: Improving the error term could be useful to show new yet unsolved conjectures. If the Riemann hypothesis is true, the prime number theorem can be strengthened considerably.
Prime gaps cannot be found , even with very good approximations of the prime-number-function. Consecutive primes are just too close. And it also remains extremely difficult to find huge primes.
Noone knows a better error term than the square-root term, which however is only valid , if the Riemann hypothesis is true. Of course, better error terms could exist.
Even extremely accurate error terms would be useless to find concrete primes, twin-primes etc. , unless we could exactly calculate the function for huge values, which almost surely is impossible. The prime number theorem is only useful for the distribution of primes. 
An often asked question is whether primes are "random". They are determined, and it can be easily decided whether any positive integer is prime or not. But the prime number theorem does not make the decision easier. Still, no efficient method to find large primes is known.
So, primes are "unpredictable", and in some sense, actually random.
A: If $\psi(x) -x= o(x^{1/2})$ then $$\frac{\zeta'(s)}{s\zeta(s)}+\frac1{s-1} =\int_1^\infty (x-\psi(x))x^{-s-1}dx$$ is analytic for $ \Re(s) > 1/2$ and $\frac{\zeta'(s)}{s\zeta(s)}+\frac1{s-1} = o(\frac1{\Re(s)-1/2})$ as $\Re(s) \to 1/2$
thus any point of $\Re(s) = 1/2$ where it extends meromorphically it will be analytic there. With $\eta(s) = (1-2^{1-s})\zeta(s)$ we know it does extend meromorphically and from the functional equation we know $\xi(1/2+it)$ is real and has some sign change around $t=14.4$, thus $\frac{\zeta'(s)}{\zeta(s)}$ has a pole on $\Re(s) = 1/2$ and hence the error term cannot be smaller than $\psi(x) -x= O(x^{1/2})$.
About why the RH is interesting : because the theory of $\zeta(s)$ leads to a huge theory at the intersection of analysis, arithmetic, algebra, geometry, with a wide range of applications in a lot of mathematical fields.
A: A few different questions have been asked here:
"is there any application to these error terms inside number theory?"
Yes. For example, RH implies that primes cannot be very far apart, and thus implies results in the direction of Cramer's conjecture.
A different context is where one wants explicit bounds on products of primes or other functions. This can show up for example in situations where one is trying to establish that one has all primes of a given form or something similar.
The truth is that the vast majority of really interesting things don't come from RH itself, but from GRH. For example, GRH implies that the deterministic Miller-Rabin primality test always work, and GRH also implies a version of Artin's primitive root conjecture.
"is there any argument to say that we cannot beat Riemann hypothesis' square root savings?"
Yes. In fact, we know that we cannot do better. Let $E(x)=Li(x)- \pi(x)$. Then  Littlewood proved in a 1914 paper the following: There is a constant c>0 such that: 
1) There are arbitrarily large values of $x$ such that $E(x) > \frac{ cx^{1/2} (\log \log x)}{\log x} $
2) There are arbitrarily large values of $x$ such that $E(x) < \frac{ -cx^{1/2} (\log \log x)}{\log x} $.
Littlewood's result means that one cannot do better than a square root x term in general except possibly by a tiny log factor. 
