REVIEW
Half-sized gaps between zeros of the Riemann zeta function
Number Theory · Submitted 19 Oct 2026 · 10 references
Pith's verdict: Clean, clever, and important: first conditional proof that normalized gaps between zeros of the Riemann zeta function can be at most 1/2, breaking a long-standing barrier with a genuinely new technique.
Why Pith says this
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Referee report
The paper
The paper claims
Assuming the Riemann hypothesis, the lower limit of normalized gaps between consecutive zeros of the zeta function is at most 1/2.
The reading
What carries the argument
The key object is the quotient function F(s) = tan(π/2 (z(s)-1/2)), where z(s) involves a logarithmic derivative integral of shifted zeta functions. The assumption that μ > 1/2 forces |F(s)| ≤ 1 + o(1) in a certain rectangle. A cleverly chosen Möbius resonator (Dirichlet polynomials R and Q with coefficients derived from μ(n) and trigonometric functions) satisfies the identity F(s)R(s) = Q(s) in the sense of Dirichlet series. By integrating F R Q against a Gaussian weight, a contradiction arises from comparing the growth of coefficient sums, proving that μ cannot exceed 1/2.
Extended reading notes
Core claim
Under the Riemann Hypothesis, the liminf of (γ_{n+1} - γ_n) log γ_n / (2π) is at most 1/2.
Load-bearing premise
The entire proof assumes the Riemann Hypothesis, that all non-trivial zeros lie on the line Re(s)=1/2.
Editorial extensions
If this is right
- If the Riemann Hypothesis holds, the maximal lower bound for gaps is reduced to 1/2, narrowing the possible range for μ.
- The method breaks the earlier barrier thought to limit the resonance technique to above 0.508, opening new approaches for gap problems.
- The technique may be adaptable to other families of L-functions to obtain similar sharp bounds.
Reading between the lines
- The proof critically depends on the Riemann Hypothesis; without it, the conclusion is not known to hold.
- The coefficient identity and the choice of resonator suggest that achieving μ < 1/2 might require an even more refined construction, likely involving longer Dirichlet polynomials or higher-order correlations.
- The success of the quotient approach indicates that other problems in value distribution of zeta functions could benefit from similar ratio-symmetric resonators.
Original abstract
Assuming the Riemann hypothesis, we prove that the lower limit of the normalised gaps between consecutive zeros of the Riemann zeta function is at most $1/2$. Our methods utilise a more efficient method for detecting gaps with short Dirichlet polynomials. Specifically, we apply resonance to a quotient of shifted zeta functions, rather than to the number of zeros in a short interval, using a M\"obius resonator adapted to the symmetry between the shifts.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Assuming the Riemann hypothesis, we prove that the lower limit of the normalised gaps between consecutive zeros of the Riemann zeta function is at most $1/2$. Our methods utilise a more efficient method for detecting gaps with short Dirichlet polynomials. Specifically, we apply resonance to a quotient of shifted zeta functions, rather than to the number of zeros in a short interval, using a M\"obius resonator adapted to the symmetry between the shifts.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Assuming the Riemann hypothesis, we prove that the lower limit of the normalised gaps between consecutive zeros of the Riemann zeta function is at most $1/2$. | result 1.00 | 0.95 | 0.96 | asserted 1.00 |
| s1 | Our methods utilise a more efficient method for detecting gaps with short Dirichlet polynomials. | method 0.93 | 0.81 | 0.02 | asserted 1.00 |
| s2 | Specifically, we apply resonance to a quotient of shifted zeta functions, rather than to the number of zeros in a short interval, using a M\"obius resonator adapted to the symmetry between the shifts. | method 0.98 | 0.91 | 0.03 | asserted 0.98 |
What was examined, and what was not
- arXiv listing abstract v1
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- relations to other papers
- P abstract
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References
Discussion
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