I'm curious about the following question:
As of 2005(?) the Riemann hypothesis is verified for the first 10 trillion zeroes, they are all on the critical line. Does this verification gives us any information about prime number?
In particular, are there any results saying if all the non-trivial zeroes whose imaginary part is < N and > 0 are on the critical line, then we understand something about prime number < M, where M is a number depend on N?
Best Answer
If you look at the explicit formula, then you can get a bound for the error term in the PNT: If $$ \psi(x) = \sum_{p^k\le x}\log p, $$ then formula (9) in page 109 of Davenport's book (multiplicative number theory) implies that $$ \psi(x) = x + \sum_{-T\le \gamma\le T}\frac{x^\rho}{\rho} + O\left(1+\frac{ x\log^2(xT) }{T}\right), $$ for every $T\ge1$, where the implied constant is completely effective. Now, if we know that all the zeroes of $\zeta$ up to height $T$ lie on the critical line, then this automatically implies that $$ |\psi(x) - x | \le x^{1/2}\sum_{-T\le \gamma\le T} \frac{1}{\sqrt{1/4+\gamma^2}} + O\left(1+ \frac{ x\log^2(xT)}{T} \right) \ll x^{1/2}\log^2T + \frac{ x\log^2(xT)}{T}, $$ for some effective implied constants. So, in certain ranges of $x$, depending on $T$, you can get very good bounds on the size of $\psi(x)$ and therefore on how many primes there are up to $x$.