Zeta Zeros on the Critical Line
Daniel A. Goldston, Ade Irma Suriajaya
TL;DR
The paper investigates how Montgomery's pair-correlation method interacts with the Riemann Hypothesis to constrain the zeros of the Riemann zeta-function. It develops a RH-free framework that analyzes horizontal pairings of zeros and separates diagonal, symmetric-diagonal, and non-symmetric contributions to derive conditional results on how many zeros lie on the critical line and are simple. It then connects these conditional RH-free bounds to the Pair Correlation Conjecture, showing that PCC implies asymptotically $100\%$ of zeros are simple and on the critical line, and that narrower zero-strip hypotheses can yield strong partial conclusions. The work highlights a deep link between pair correlation statistics and the vertical distribution of zeros, with implications for prime number distribution and unconditional or conditional zero-density results.
Abstract
Montgomery in 1973 introduced the pair correlation method to study the vertical distribution of Riemann zeta-function zeros. This work assumed the Riemann Hypothesis (RH). One striking application was a short proof that at least 2/3 of zeta-zeros are simple zeros, the first result of its type. Over the last 50 years, most work on pair correlation of zeta-zeros has continued to assume RH. Here we show that if RH could be removed from Montgomery's simple zero proof, then this would also give a proof that 2/3 of the zeros are on the critical line. While at present we cannot unconditionally obtain this result, we have proved in joint work that this is true if we assume all the zeros are in a narrow enough region centered on the critical line. In separate joint work we use the same idea with a related pair correlation method to prove that assuming the Pair Correlation Conjecture, also without the assumption of RH, we obtain 100\% of the zeros are simple and on the critical line.