# 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!

• From the practical perspective, prime numbers are used in cryptography, e.g. RSA. Large prime numbers are essential. Due to PNT we have this approximation of prime numbers $$p_n \sim n\log{n}$$ But, because of the large error term for this approximation, finding the next largest prime number using it is still impractical. – rtybase Nov 22 '18 at 22:33