The pair correlation method of zeros of the Riemann zeta-function and the Alternative Hypothesis, without RHMPIM

by Ade Irma Suriajaya, Junghun Lee (Kyushu University/MPIM)

Thursday Sep 11, 2025, 3:00 PM → 4:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Number theory lunch seminar

Montgomery (1973) suggested an approach to study the pair correlation of nontrivial zeros of the Riemann zeta-function, and proved the corresponding asymptotic formula within a limited range assuming the Riemann Hypothesis (RH). The extended behavior remains a conjecture which implies the famous Pair Correlation Conjecture (PCC) for these zeros. In Suriajaya's previous work with Siegfred Alan C. Baluyot, Daniel Alan Goldston, and Caroline L. Turnage-Butterbaugh, they have showed how to remove RH in such pair correlation methods and recover Montgomery results on the proportion of simple zeros under hypotheses weaker than RH. They have in addition obtained the proportion of zeros lying on the critical line, which we simply call critical zeros for brevity.
    In a follow-up work with Daniel Goldston and Jordan Schettler, we showed that PCC without RH implies that asymptotically 100% of the zeros are simple and critical, thus RH is asymptotically true. Our method applies to other pair correlation conjectures as well, such as the Alternative Hypothesis (AH). AH first arose as a consequence of the possible existence of Landau-Siegel zeros of real Dirichlet L-functions, as demonstrated by D. R. Heath-Brown at a 1996 AIM conference. In this talk we would like to introduce the main ideas behind our results.