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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!

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    $\begingroup$ 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. $\endgroup$ – rtybase Nov 22 '18 at 22:33
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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.

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