Zero-density exponent for ζ beyond Guth–Maynard: N(σ,T) ≤ T^(A(σ)(1−σ)+o(1)) with A < 30/13
One improvement in five decades, now with a new method that others are already extending, so 61: research-grade, just below Kakeya in ℝ⁴ (62) and below the Lindelöf exponent (74), which needs pointwise rather than averaged control. Application value 14 reflects explicit gains for primes in short intervals and least primes in progressions. Leverage 44: many prime-distribution theorems take the zero-density exponent as input.
The problem
Show that very few zeta zeros can lie off the critical line, with a better count than the 2024 record.
- 1Zeros off the line would hurt prime counts
- 2Count them: few, with a power-law bound
- 3Record bound 12/5 falls to 30/13 in 2024
- 4Next: lower still, or reach the exponent 2
Setup. Let count zeros of with and . For let be the least exponent with The asserts for all . It would follow from the , and is implied by the Riemann Hypothesis (which says for ). The point of the exponent: for near , the bound says that zeros far from the line are very rare.
Built from
earlier problems → this oneThe problems this one is built from: progress or tools it needs (builds on), special cases that come first (generalizes), and weaker results on the way (implies). On the right, what builds on it.
- Start with
- Zero-density exponent for ζ beyond Guth–Maynard: N(σ,T) ≤ T^(A(σ)(1−σ)+o(1)) with A < 30/13This problem
- Then
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Harmonic analysisTheory
- Step 2
- Step 3
- The Riemann zeta function and prime distributionTheory · host
- Then
- Zero-density exponent for ζ beyond Guth–Maynard: N(σ,T) ≤ T^(A(σ)(1−σ)+o(1)) with A < 30/13The problem
State of the art
Open. Ingham (1940) gave A(σ) ≤ 3/(2−σ) and Huxley (1972) A(σ) ≤ 3/(3σ−1) for σ ≥ 3/4, so sup A ≤ 12/5, attained at σ = 3/4. Guth and Maynard (arXiv:2405.20552, 2024) proved new large-value estimates for Dirichlet polynomials of length N taking values near N^(3/4), giving A(σ) ≤ 15/(3+5σ) and sup A ≤ 30/13, the first improvement of 12/5 in five decades; they also obtain prime asymptotics in intervals of length x^(17/30+o(1)). Chen, Gupta and Li (arXiv:2507.08296) adapted the method to the sum over characters mod q, with exponent 7/3 in place of Huxley's 12/5. Tao, Trudgian and Yang (arXiv:2501.16779) built an exponent database and obtained new zero-density estimates in several ranges by systematic optimization. The density hypothesis is known for σ ≥ 25/32 (Bourgain, 2000). A zero-free half-plane Re s > 7/8, as claimed in an unrefereed OpenAI manuscript (October 2026), would make N(σ,T) = 0 for σ > 7/8 but would not change the supremum, which is attained near σ = 7/10. No refereed improvement of 30/13 for ζ was found as of October 2026.
Realistic applications
Zero-density estimates give primes in short intervals, Linnik-type bounds on least primes in progressions, and Goldbach-type results on almost all even integers; each exponent improvement lowers the thresholds in those theorems.
What counts as solved
A proved bound N(σ,T) ≪ T^(A(σ)(1−σ)+o(1)) with sup A < 30/13 or the density hypothesis in a wider range, with full proof refereed.
How it gets verified
Expert reviewOnly expert review of the argument is possible today.
The target is a bound N(sigma,T) <= T^(A(sigma)(1-sigma)+o(1)) with sup A < 30/13. Mathlib defines riemannZeta but has no zero-counting or large-values machinery, so a Lean check is not realistic today. The exponent optimisation can be reproduced numerically, but resolution needs a refereed full proof.
Claims under verification
+ add a claimNo claimed results on the Verification board for this problem yet.
Where the work is
Mostly new analytic ideas; computer optimization of exponent combinations helps.
Bottlenecks
ranked by votes- #1TheoryLarge-value estimates at the intermediate range
Guth–Maynard improve large values only for Dirichlet polynomials taking values near N^(3/4); the new bound loses to Ingham below σ = 7/10, where the supremum now sits, and no mechanism reaches the density hypothesis there.
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Connected problems
Progress
- Jul 11, 2025Extension to Dirichlet L-functionssource
Chen–Gupta–Li obtain exponent 7/3 for the sum over characters mod q.
- May 31, 2024Guth–Maynard: 30/13source
A(σ) ≤ 15/(3+5σ); with Ingham this gives sup A ≤ 30/13, the first improvement of Huxley’s 12/5.
- Jan 1, 1940Ingham's zero-density boundsource
A(σ) ≤ 3/(2−σ) (date approximate to the year; cited in Guth–Maynard).