What question did this study set out to answer?

To determine the proportion of zeros of Dirichlet L-functions that are simple and lie on the critical line at polylogarithmic height when averaged over non-principal characters.

August 19, 2026Open Access

Two thirds of the zeros of Dirichlet L-functions at polylogarithmic height are simple and on the critical line, on average

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To determine the proportion of zeros of Dirichlet L-functions that are simple and lie on the critical line at polylogarithmic height when averaged over non-principal characters.
Analyzed Dirichlet L-functions L(s, χ) within height windows (T, 2T] for (log q)^(2+ε) ≤ T ≤ (log q)^A, averaged over q−2 non-principal characters modulo large primes q.
Implemented bilinear-sums accounting with explicit constants alongside Montgomery–Taylor window variants.
Formally machine-checked 70 supporting mathematical lemmas using the Lean 4 proof assistant.
Established unconditionally that at least 2/3 − O(1/log log q) of the zeros in the specified polylogarithmic window are simple and lie on the critical line.
Achieved proportion bounds of 0.6725007 and 0.8362503 using Montgomery–Taylor window variants with profile-robust Gevrey base constants.

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Implication

Theoretical analysis proves that at least two-thirds of Dirichlet L-function zeros are simple and on the critical line, advancing understanding of the Generalized Riemann Hypothesis.

Cite This Study

Fredrik Prüzelius (2026) studied this question.

synapsesocial.com/papers/6a85638803308d306e2d6c24 https://doi.org/10.5281/zenodo.21986495

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Implication

Theoretical analysis proves that at least two-thirds of Dirichlet L-function zeros are simple and on the critical line, advancing understanding of the Generalized Riemann Hypothesis.

Two thirds of the zeros of Dirichlet L-functions at polylogarithmic height are simple and on the critical line, on average

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Discussion

Implication

Cite This Study

Two thirds of the zeros of Dirichlet L-functions at polylogarithmic height are simple and on the critical line, on average

Key Points

Discussion

Implication

Cite This Study

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