Work

Pair correlation conjecture for the zeros of the Riemann zeta‐function I: Simple and critical zeros

Year: 2026

Type: article

Abstract: Montgomery in 1973 introduced the pair correlation conjecture (PCC) for zeros of the Riemann zeta‐function. He also conjectured that asymptotically 100% of the zeros are simple. His reasoning to support these two conjectures used the Riemann hypothesis (RH). Building on Montgomery's approach, Gallagher and Mueller proved in 1978 that PCC under RH implies that 100% of the zeros are simple. Actually, the method of Gallagher and Mueller does not depend on RH, and thus Montgomery's second simplicity conjecture follows from his PCC conjecture without additionally assuming RH. We clarify this result by explicitly not assuming RH and considering PCC as a conjecture only concerning the vertical distribution of zeros. We then show that, for the first time, PCC can also be used to obtain information on the horizontal distribution of zeros. Using Gallagher and Mueller's method and a new idea concerning “horizontal multiplicity,” we use PCC to prove that asymptotically 100% of the zeros are not only simple but also on the critical line.

Source: Mathematika

DOI: 10.1112/mtk.70127

Authors: Daniel A. Goldston, Junghun Lee, Jordan Schettler, Ade Irma Suriajaya

Institutions: San Jose State University, Chonnam National University, Kyushu University

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze