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Determine the proportion of low-lying zeros of Dirichlet L-functions that are simple and lie on the critical line when averaged over non-principal characters modulo a large prime.

August 19, 2026Open Access

Two thirds of the low-lying zeros of Dirichlet L-functions, on average

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Determine the proportion of low-lying zeros of Dirichlet L-functions that are simple and lie on the critical line when averaged over non-principal characters modulo a large prime.
Derived an unconditional proof for family-averaged Dirichlet L-functions over q - 2 non-principal characters modulo a sufficiently large prime q in the window T ≍ (log q)^(2+ε).
Formally verified 70 supporting mathematical lemmas using the Lean 4 interactive theorem prover.
Established unconditionally that at least 2/3 − O(1/log log q) of the low-lying zeros of L(s, χ) are simple and lie on the critical line.
Demonstrated the validity of this proportion within the family-averaged polylog-window regime.

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Implication

Mathematical analysis demonstrates that two-thirds of low-lying zeros in Dirichlet L-functions lie on the critical line, indicating major progress toward the Generalized Riemann Hypothesis.

Cite This Study

Fredrik Prüzelius (2026) studied this question.

synapsesocial.com/papers/6a858aaf03308d306e2d7f19 https://doi.org/10.5281/zenodo.21980225

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Implication

Mathematical analysis demonstrates that two-thirds of low-lying zeros in Dirichlet L-functions lie on the critical line, indicating major progress toward the Generalized Riemann Hypothesis.

Two thirds of the low-lying zeros of Dirichlet L-functions, on average

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Discussion

Implication

Cite This Study

Two thirds of the low-lying zeros of Dirichlet L-functions, on average

Key Points

Discussion

Implication

Cite This Study

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