Small and large gaps between zeros of Dirichlet $L$-functions

Andrew Pearce-Crump; Benjamin Durkan

REVIEW

Small and large gaps between zeros of Dirichlet $L$-functions

Benjamin Durkan, Andrew Pearce-Crump

Number Theory · Submitted 20 Oct 2026

Pith's verdict: Theorem 1 is a genuinely new unconditional half-spacing result for Dirichlet L-functions and the method looks sound; the large-gap constant needs a shipped certificate before the paper is fully verifiable.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on October 21, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 3 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2610.28527 (version 2)

Pith IDGF4H5VXZ, Pith's permanent record of this paper and its review

CitationAndrew Pearce-Crump, Benjamin Durkan. "Small and large gaps between zeros of Dirichlet $L$-functions." arXiv:2610.28527 (2026). https://pith.science/paper/GF4H5VXZ

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Chief thing to know: this is a serious paper with a genuinely new small-gap method. Theorem 1 — mu <= 1/2 unconditionally for every fixed primitive Dirichlet L-function, zeta included — is the headline. It goes past the Montgomery-Odlyzko barrier of 0.5042 in a natural setting, and the moment/density argument behind it looks coherent. The catch: the large-gap constant 3.63846723 rests on a finite computation that is described but not shipped, so the manuscript as posted cannot be fully checked.

What is actually new: the unconditional half-spacing theorem for all fixed primitive characters; a positive-proportion strengthening for zeta under RH; and a large-gap constant that beats the earlier 3.18 under RH. The method itself is the interesting part. The authors compute all fixed moments of a shifted logarithmic difference, identify those moments with a positive measure on two horizontal lines, and then select a test function whose density is negative. Lemma 3 and Lemma 4 look like the right machinery for that passage, and the off-critical-line dichotomy is handled correctly, including complex characters. That is a real new route around the old criteria, not a repackaging.

Theorem 2 is plausible and I did not find an obvious hole in Section 7. The authors themselves flag that the counted spacings may include zero-length gaps and that c_epsilon may tend to zero; those statements are accurate and should not be read as defects.

The soft spot is exactly where the stress-test puts it: Section 8.2. Inequality (8.4) is the load-bearing step for Theorem 3, the margin is about 1.08e-9, and the 1638-coefficient polynomial H is neither listed nor shipped. The aggregate interval bounds and coefficient mass do not let a reader reconstruct H. A single transcription error in a monomial could change the sign. This is a reproducibility gap, not a demonstrated inconsistency: Theorems 1 and 2, and the analytic structure of Theorem 3, do not depend on it. The fix is easy — release the coefficient vector and the Arb code, or a machine-checkable certificate — and I would make that a condition of acceptance, not grounds for rejection.

Who this is for: analytic number theorists working on zeros of L-functions. The paper deserves a serious referee even though my own verdict is conditional. Send it to review, with the certificate requirement attached.

What would settle it

Re-run the finite generalised-eigenvalue maximisation of $J_{32}/I$ at $\ell = 39327/20000$ and $\phi = 3.63846723$ over the 1638 monomials of Section 8.2, using the paper's formulas for the moments (8.2), the Taylor expansion of the sine, and the remainder bound (8.3): if no rational coefficient vector with denominators dividing 1080 and total coefficient mass below 7.46135 makes $\frac{4\ell}{\pi}J(\phi) - \phi I > 0$, Theorem 3's constant is not established. A complementary check of Theorem 1 is to compare, at large $T$ for a fixed small conductor, direct weighted averages of $V_\phi^j$ with the limiting values $\int_0^1 (A_\phi^j f)(x)f(x)\,dx\,/\,\|f\|_2^2$; a persistent mismatch at any fixed $j$ would signal a flaw in the moment passage.

Referee report

Significance

If Theorems 1 and 3 are accepted, the paper is a substantial advance: Theorem 1 reaches half the mean spacing unconditionally for all fixed primitive Dirichlet L-functions, and Theorem 3 substantially improves the known large-gap constant under RH. The analytic machinery is presented in considerable detail: Propositions 1 and 2 give explicit moment formulas, Lemma 3 gives a clean uniqueness and density statement, and the restriction to squarefree Möbius coefficients is carried through carefully. The authors are also explicit about the limitation of Theorem 2, which counts zero-length spacings and does not give a positive proportion at or below half the mean spacing. The central weakness is that the decisive numerical input behind Theorem 3 is not reproducible from the manuscript; the strict inequality (8.4) has a margin of about 1.08 × 10^{-9} and rests on reported interval-arithmetic outputs rather than on a listed polynomial or shipped code. This verification gap currently blocks acceptance, but it is fixable by providing a certificate.

Major objections (2)

[Section 8.2 (proof of Theorem 3)] The decisive inequality (8.4) is not independently checkable from the manuscript. The proof reduces Theorem 3 to the existence of a real polynomial H in a 1638-dimensional space; the paper gives the space, the quadratic-form formulas for I and J_K, the parameter values ell = 39327/20000 and phi = 3.63846723, and aggregate interval enclosures for 10^34 I and 10^34 J_32 together with the remainder bound (8.3). It does not give the coefficient vector b, the generalized-eigenvalue computation, or the interval-arithmetic script, and the claimed strict margin is about 1.08 × 10^{-9}. Since an error in a single coefficient or in one interval evaluation could change the sign of (8.4), Theorem 3 rests on an unverifiable finite computation in this version. A supplementary file with the exact polynomial H (or the coefficient vector) and a verifiable certificate or evaluation code is needed; without it the numerical constant cannot be accepted.
[Section 8.2, formula for J_K(phi)] As printed, the displayed formula for J_K(phi) appears to multiply three Gamma factors: Gamma(ell^2 + r + 5l_1/4 + 3l_2/2 + 2l_3 + 3l_4), Gamma(ell^2 + r + 5k_1/4 + 3k_2/2 + 2k_3 + 3k_4 + 2k + 1), and Gamma(2k + 5/4(k_1 - l_1) + 3/2(k_2 - l_2) + 2(k_3 - l_3) + 3(k_4 - l_4) + 1). The sentence immediately before says that the gamma factors follow from the beta integral in y, which instead requires division by the second of these factors, namely Gamma(A + B) in Gamma(A)Gamma(B)/Gamma(A + B). If the printed product is what was evaluated, the values of J_32 and hence the inequality (8.4) would change; if the missing division is a typesetting omission, it should be corrected. Please confirm which expression was used in the interval computation.

Minor objections (3)

[Section 8.2] The sentence introducing the Taylor truncation says "through degree 2K + 2", but the displayed sine polynomial contains odd powers through degree 2K + 1 and the remainder bound (8.3) has exponent 2K + 3; the wording should be corrected to "through degree 2K + 1".
[Section 6.1, near (6.4)] In the prose explanation of the imaginary-part identity, the conjugate bars appear to be missing from h_0 and C h_0; as rendered, the sentence does not state the identity correctly. Please insert the overlines consistently.
[Theorem 2 statement] The caveat that the count includes spacings of length zero is stated only after the theorem; moving it into the statement itself, or at least into the abstract, would prevent misreading the result as a positive proportion of strictly positive gaps at or below half the mean.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Unconditionally, every fixed Dirichlet L-function has infinitely many consecutive zero gaps below $(1/2+\varepsilon)$ times the mean; under RH, zeta gets a positive proportion of such gaps and large gaps exceed $3.63846723$ times the mean.

The reading

The zeros of Dirichlet L-functions encode how primes are distributed in arithmetic progressions, and the gaps between consecutive zeros are among the least controllable features of that record. This paper proves that for every fixed primitive Dirichlet L-function — the Riemann zeta function included — infinitely many consecutive zeros lie closer together than $(1/2+\varepsilon)$ times the average spacing, for every $\varepsilon>0$, with no unproved assumption; all earlier numerical small-gap bounds sat strictly above $1/2$ and mostly assumed the Riemann Hypothesis. The engine is a moment calculation: count zeros in a moving window as the imaginary part of a shifted logarithmic difference of the completed L-function, compute every fixed moment of that variable under a Möbius-squared weight, and show that the resulting moments force a density that is negative somewhere — impossible for a positive measure. Under the Riemann Hypothesis the same machinery gives a positive proportion of such small gaps for zeta and, for every fixed primitive character, infinitely many consecutive gaps larger than $3.63846723$ times the mean — a new record.

What carries the argument

The load-bearing object is the operator $(A_\phi f)(x) = i\phi f(x) + \frac{2i}{\pi}\int_0^x \frac{\sin(\pi\phi u)}{u} f(x-u)\,du$ on continuous functions on $[0,1]$, obtained as the $T\to\infty$ limit of the action of prime multiplication on the Möbius-weighted Dirichlet polynomial; its $j$-th power gives the $j$-th weighted moment of the zero-counting variable $V_\phi$. The two facts that make it useful are that its spectrum is the single point $\{i\phi\}$ and that its exponential $e^{\pi A_\phi}$ acts on the exponential $e^{-i\pi\phi x}$ by a simple factor, so the density formula of Lemma 3 becomes explicit. The choice $f(x) = e^{-i\pi\phi x} - e^{i\pi\phi}e^{i\pi\phi x}$ cancels the inverse appearing in that formula and leaves the density proportional to $\sin(\pi\phi)\sin(2\pi\phi)$, whose sign change at $\phi = 1/2$ is exactly the half-spacing barrier. For large gaps the machinery is different: coefficients indexed by reduced fractions $m/n$ with symmetric dependence on the numerator and denominator, two explicit quadratic forms $I$ and $J(\phi)$ built from moments of a four-dimensional limiting measure of prime-factor size sums, and the $1638$-coefficient polynomial $H$ chosen to make $(1-\phi)I + (4\ell/\pi)J(\phi) > 0$.

Extended reading notes

Core claim

The paper's central claim is that one mechanism reaches two classical gap constants. Theorem 1 states that $\mu \le 1/2$ unconditionally for every fixed primitive Dirichlet character $\chi$, meaning that for every $\varepsilon>0$ infinitely many consecutive normalised gaps fall below $1/2+\varepsilon$. Off-line zeros occur in pairs with identical ordinates, so infinitely many of them would already force $\mu=0$; otherwise all sufficiently high zeros lie on the critical line and the proof below applies in full. The argument assumes $\mu>1/2$ and derives a contradiction from the moments of the zero-counting logarithmic difference $V_\phi(t)$, whose imaginary part counts zeros in an interval of length $2\pi\phi/\log T$. Under the gap assumption this variable takes only values on two horizontal lines, so its weighted moments — evaluated exactly for every fixed order by contour integration against the squared modulus of a Möbius Dirichlet polynomial — must be the moments of a positive measure. Lemma 3 turns those moments into explicit densities on the two lines; for the test function $f(x)=e^{-i\pi\phi x}-e^{i\pi\phi}e^{i\pi\phi x}$ the lower-line density at zero equals $\sin(\pi\phi)\sin(2\pi\phi)/(\pi\phi\|f\|_2^2)$, which is negative for every $1/2<\phi<1$. A positive measure cannot have negative density, so $\mu\le 1/2$ follows. Theorem 2, assuming RH for zeta, keeps the same sign mechanism on a shortened interval $[0,\theta]$ and turns the contradiction into a counting argument that yields $\gg T\log T$ small spacings. Theorem 3, assuming RH for each fixed character, makes the weighted mean number of zeros in an interval less than one, which forces an empty interval; symmetric coefficients on reduced fractions $m/n$ reduce that mean to a ratio of two quadratic forms, and a $1638$-coefficient polynomial — found by a finite generalised-eigenvalue optimisation and certified by interval arithmetic — makes the ratio less than one at $\phi=3.63846723$.

Load-bearing premise

Theorems 2 and 3 explicitly assume the Riemann Hypothesis, and Theorem 3's constant rests on a $1638$-coefficient polynomial $H$ that the manuscript never lists, reporting only aggregate interval-arithmetic bounds on its coefficients, so the inequality (8.4) behind $3.63846723$ cannot be checked from the paper alone; Theorem 1 instead stands on its proven moment limits and the pointwise density evaluation.

Editorial extensions

If this is right

For every fixed primitive Dirichlet character, infinitely many consecutive zero gaps are smaller than $1/2+\varepsilon$ times the mean spacing, for every $\varepsilon>0$, with no unproved assumption.
If infinitely many zeros lie off the critical line at positive heights, their equal-ordinate pairs already give $\mu=0$; otherwise all high zeros are on the line and Theorem 1's proof applies, so the small-gap conclusion is unconditional either way.
Under RH for zeta, at least $c_\varepsilon T\log T$ consecutive gaps below height $2T$ are shorter than $1/2+\varepsilon$ times the mean; the count includes zero-length spacings from off-line pairs, and $c_\varepsilon$ may tend to zero as $\varepsilon\to 0$, so no positive proportion is claimed at the half-spacing mark itself.
Under RH for a fixed primitive character, infinitely many consecutive normalized gaps exceed $3.63846723$ times the mean spacing.
The constant $1/2$ is exactly the first positive half-integer on the Alternative Hypothesis picture, and the paper notes that Theorems 1–3 remain compatible with it.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the moment method consumes all fixed moments rather than a single weighted mean, it clears the known obstruction that no coefficient choice in the classical weighted-mean method reaches below $0.5042$ for length-$T^{1-\delta}$ polynomials; extending the same all-moments strategy to other coefficient families is the most direct route to push the large-gap constant higher.
Making the 1638-coefficient vector public would turn the numerical part of Theorem 3 into a reproducible computation and would let others test whether larger polynomial spaces, different prime-weight exponents, or extra prime sums raise the constant further.
The two-line measure structure — moments of a logarithmic difference against a multiplicative weight, with density recovered by an exponential substitution — is generic enough that the same density-negativity test should transfer to other families of $L$-functions wherever the analogous prime sums can be evaluated.

Original abstract

For every fixed primitive Dirichlet $L$-function and every $\varepsilon>0$, we prove unconditionally that infinitely many consecutive zeros have spacing less than $1/2+\varepsilon$ times the mean. Assuming the Riemann Hypothesis, we also prove a positive proportion of such spacings for the Riemann zeta-function, and infinitely many spacings greater than $3.63$ times the mean for every fixed primitive Dirichlet $L$-function.

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.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-21: 1 request, 2,409 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.55 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.15 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

For every fixed primitive Dirichlet $L$-function and every $\varepsilon>0$, we prove unconditionally that infinitely many consecutive zeros have spacing less than $1/2+\varepsilon$ times the mean. Assuming the Riemann Hypothesis, we also prove a positive proportion of such spacings for the Riemann zeta-function, and infinitely many spacings greater than $3.63$ times the mean for every fixed primitive Dirichlet $L$-function.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	For every fixed primitive Dirichlet $L$-function and every $\varepsilon>0$, we prove unconditionally that infinitely many consecutive zeros have spacing less than $1/2+\varepsilon$ times the mean.	result 0.98	0.95	0.67	asserted 1.00
s1	Assuming the Riemann Hypothesis, we also prove a positive proportion of such spacings for the Riemann zeta-function, and infinitely many spacings greater than $3.63$ times the mean for every fixed primitive Dirichlet $L$-function.	result 0.99	0.93	0.19	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.