The prime number theorem and pair correlation of zeros of the Riemann zeta-function - Goldston, D. A.; Suriajaya, Ade Irma (Research in number theory, 2022)
We prove that the error in the prime number theorem can be quantitatively improved beyond the Riemann Hypothesis bound by using versions of Montgomery’s conjecture for the pair correlation of zeros of the Riemann zeta-function which are uniform in long ranges and with suitable error terms.