Zeta Zeros on the Critical Line

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 simple and on the critical line. This idea has been applied in several recent papers to obtain other results on the zeros.

💡 Research Summary

The paper revisits Montgomery’s 1973 result that at least two‑thirds of the non‑trivial zeros of the Riemann zeta‑function are simple, a theorem that historically required the Riemann Hypothesis (RH). The authors show that the same proportion can be obtained without assuming RH, and in fact the same argument yields that at least two‑thirds of the zeros are simultaneously simple and lie on the critical line σ = ½.

The key technical innovation is to treat the set of zeros as a multiset: each zero ρ = β + iγ is counted with multiplicity m_ρ, and the total number of zeros up to height T is N(T)=|Z(T)| while the weighted count is N⁎(T)=∑_{ρ∈Z(T)} m_ρ. This allows the authors to keep track of repeated zeros and of “horizontal” coincidences (different zeros sharing the same ordinate γ).

The classical Montgomery argument uses the Fejér kernel sum

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