Zero density in short strips and short-interval bounds toward Hypothesis S

Yangbo Ye

REVIEW

Zero density in short strips and short-interval bounds toward Hypothesis S

Yangbo Ye

Number Theory · Submitted 11 Oct 2026

Pith's verdict: Clean technical extension of Guth–Maynard to short strips; honest about its limits, but the advance is narrow.

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arXiv ID2610.15989 (version 2)

Pith IDPO4U3RMD, Pith's permanent record of this paper and its review

CitationYangbo Ye. "Zero density in short strips and short-interval bounds toward Hypothesis S." arXiv:2610.15989 (2026). https://pith.science/paper/PO4U3RMD

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Editor's note

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The paper extends the Guth–Maynard large-value estimates and Maynard–Pratt Type II zero argument to short vertical strips, producing explicit zero-density exponents for the Riemann zeta function on the range 7/10 ≤ σ ≤ 4/5. The main result (Theorem 1.2) is a genuine improvement over Ye–Zhang on that interval—at σ = 3/4 the exponent drops from 3a/5 to 5a/9. The paper then plugs these density bounds into an explicit formula to obtain power savings for smoothed prime-power sums with phase e(αn^β). The zero-density argument is the real contribution; the prime-sum applications are natural but not surprising given the known density input.

What the paper does well: it writes out the adaptation cleanly, handles the localization and the frequency dependence explicitly, and is consistently honest about limitations. The authors state flatly that the method does not improve cusp-form density bounds even when temperedness is known, and that the prime-sum applications for ordinary primes give weaker ranges than existing direct estimates (Matomäki–Shao–Tao–Teräväinen, etc.). No overclaiming.

Soft spots in proportion: (1) The improvement is narrow—only on 7/10 < σ < 4/5, and the exponent gain is modest. (2) The load-bearing short-interval fourth moment of ζ (Heath-Brown, Iwaniec) is cited but not re-proved, which is fine for a research paper but means the result is conditional on that deep input. (3) The cusp-form applications (Theorems 1.4–1.5) use earlier density bounds from Ye–Zhang, so they do not benefit from the new zeta estimate; the paper acknowledges this. (4) The machinery is heavy: the argument depends on Guth–Maynard, which is already a long paper. A reader wanting to verify all steps would need to cross-reference several sources.

Overall the paper is technically sound, well-organized, and honest. It delivers a real but narrow improvement. The stress-test found no internal inconsistency or misuse of citations.

Recommendation: This paper deserves a serious referee. The referee should check the adaptation of the Type I argument (the dyadic grouping and the choice of Z) and the handling of the frequency uniformity in the applications, but I see no reason to suspect a fatal flaw. The paper is suitable for a number-theory journal.

Referee report

Major objections (2)

[Section 3, after Proposition 3.3] The argument for uniformity in the varying frequency $m$ (the mesh approximation) is described only in a few sentences. The text says "enlarge $a$ to the next mesh point and use a bounded covering" and then "choose the mesh and the density loss sufficiently small in terms of $\varepsilon/(1+\beta+R)$". This is too vague to be verified. The paper should provide an explicit construction of the mesh (how many points, the step size, how the covering is achieved) and a clear justification that the total loss is $O(K^\varepsilon)$ rather than $O(K^c)$ for some $c>0$. Since the uniformity assertion is load-bearing for Theorems 1.3–1.5, this step needs a more rigorous treatment.
[Section 2, Lemma 2.1 (Type II bound)] In the proof, the integration in (2.7) is over $[T-H-V, T+H+V]$, which is longer than $O(H)$. The paper asserts that $V = o(H)$ and that this interval can be covered by $O(1)$ intervals of the form $[U, U+U^a]$ with $U\asymp T$, using the moment hypothesis (1.1) in each such interval. For $a=2/3$, $H = T^{2/3}$, and $V = (\log T)^2$, so $V = o(H)$ holds, and the covering claim is plausible. Nevertheless, the covering should be made explicit: the interval can be written as a union of at most $\lceil 2H/(T^a) \rceil + 1$ subintervals (which is $O(1)$ because $T^a \gg H$). The current text is too terse for a load-bearing lemma.

Minor objections (5)

[Theorem 1.2] The range for $\sigma$ is given as $1/2\le \sigma <1$, but the proof in Section 2 only derives the exponent for $3/5\le \sigma <1$; the branches for $\sigma<3/5$ are taken from Ye–Zhang. The text should explicitly note that the improvement applies only for $\sigma\ge 7/10$ (already stated), but also that the branch $B(\sigma)$ for $\sigma\in[1/2, 7/10]$ is not new.
[Equation (2.15)] The constant $250$ appears without explanation. It seems to be derived from earlier parameters, but a brief calculation or reference would be helpful.
[Lemma 2.2 (Type I zeros)] The argument that all extraction and coefficient-normalisation costs are absorbed into $T^\varepsilon$ is acceptable, but a note on why the number of groups is $O(\log T)$ and why the normalisation factor $C$ is bounded by $T^{\eta/2}$ would improve readability.
[Propositions 3.1–3.2] The weighted stationary phase theorem [13, Theorem 1.1] is applied with parameters $M=N=2$ and $M=N=4$. For $\beta=1$ the second derivative $F''$ is identically zero, and the paper claims this is allowed. Please confirm that the theorem indeed permits $F''\equiv 0$ (e.g., by an explicit remark or a separate limiting argument).
[Nowhere else] In the statement of Theorem 1.3, the constant $\nu_m = \max\{\lambda + \log m/\log K,\, 2(\beta+\log m/\log K)/3\}$ is used. A brief justification that this equals $\max\{v, 2b/3\}$ as later noted would help.

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The paper

The paper claims

Improved zero-density estimates for the Riemann zeta function in short vertical strips, and applications to prime exponential sums with phase α n^β.

The reading

This paper improves the zero-density estimates for the Riemann zeta function in short vertical strips (intervals of length T^a on the imaginary axis). The main result gives a better exponent for the number of zeros with real part > σ when σ lies between 7/10 and 4/5, extending the recent Guth–Maynard large-value method to short intervals. Via the explicit formula, these density bounds yield power-saving estimates for certain exponential sums over primes weighted by a smoothed short interval, making concrete progress toward Iwaniec–Luo–Sarnak's Hypothesis S. The method also applies to automorphic L-functions, though with no improvement over existing density bounds for cusp forms given current moment inputs.

What carries the argument

The large-value estimate of Guth and Maynard for Dirichlet polynomials and the Type II zero integral argument of Maynard and Pratt, adapted to short strips via a detector function (a mollified Dirichlet polynomial combined with a gamma factor integral). This detector separates zeros into Type I (controlled by the large-value estimate) and Type II (controlled by moment bounds). The short-interval fourth moment of the Riemann zeta function (cited from Heath-Brown and Iwaniec) supplies the moment input for Type II. For the automorphic applications, the earlier Ye–Zhang density bounds for cusp forms are used instead of any new short-strip density estimate.

Extended reading notes

Core claim

The paper establishes a general moment-to-density theorem (Theorem 1.1) for globally tempered automorphic L-functions on short strips. Applied to the Riemann zeta function using the known short-interval fourth moment (Heath-Brown, Iwaniec), it obtains N_ζ(σ; T, T+T^a) ≪ $T^{{a B(σ)+ε}}$ with B(σ)=15(1-σ)/(3+5σ) for 7/10 ≤ σ ≤ 4/5 and 2/3 ≤ a ≤ 1. This improves the earlier exponent of Ye–Zhang on that σ-range and localizes the Guth–Maynard full-interval bound to intervals of length T^a. The paper then derives explicit power savings for smoothed prime-power sums with phase exp(2π i α n^β) for fixed α≠0, 0<β<1, using the explicit formula. The minimal interval length thresholds for obtaining a power saving are 2/3 (for zeta), 3/4 (for GL(2) cusp forms twisted by a real character), and 5/6 (for self-dual GL(3) Maass cusp forms). Uniform frequency dependence for mα with 1 ≤ m ≤ K^R is also given.

Load-bearing premise

The short-interval fourth moment of the Riemann zeta function, ∫$_T^{{T+T^a}}$ |ζ(1/2+it)|^4 dt ≪ $T^{{a+η}}$ for a ≥ 2/3, is cited from Heath-Brown and Iwaniec and not proved in the paper; if this bound were false or required a larger minimal a, the main zero-density improvement for zeta would be lost.

Editorial extensions

If this is right

For the Riemann zeta function, the zero-density exponent on 7/10 < σ < 4/5 is strictly improved, narrowing the gap toward the density hypothesis.
The smoothed prime-power sum S_ζ(mα, β; K, K^δ) admits a power saving over the interval length K^δ whenever δ > d(β), with d(β) as low as 2/3 (at β=8/9), and the saving is effective uniformly for m up to K^R.
For twists of self-dual GL(2) cusp forms by a real primitive character, a power saving holds for β < 2/3 with minimal interval threshold 3/4, without assuming the Ramanujan conjecture.
For self-dual GL(3) Maass cusp forms, a power saving holds for β < 1/2 with minimal interval threshold 5/6, also without Ramanujan.

Reading between the lines

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

If improved short-interval moments for cusp-form L-functions become available (e.g., via progress on the Ramanujan conjecture or subconvexity), the same framework could yield new zero-density bounds for those L-functions and extend the power-saving range.
The fact that the improvement is limited to the strip 7/10 < σ < 4/5 suggests that a different mechanism would be needed to push beyond these endpoints, perhaps requiring new large-value estimates or moment inputs.
The technique of using the explicit formula to convert short-strip zero-density estimates into bounds on exponential sums over primes is general and could be applied to other L-functions once corresponding density estimates are obtained.

Original abstract

We deduce zero-density estimates in short vertical strips from short-interval moments of globally tempered automorphic $L$-functions, by adapting the large-value estimate of Guth and Maynard and the Type II zero argument of Maynard and Pratt to the framework of Ye and Zhang. For the Riemann zeta function, we obtain $N_\zeta(\sigma;T,T+T^a)\ll T^{15a(1-\sigma)/(3+5\sigma)+\varepsilon}$ for $2/3\le a\le1$ and $7/10\le\sigma\le4/5$, improving Ye--Zhang's exponent in the interior of this range. Combining the localized density bound with the recent quasi-Riemann theorems, we estimate smoothed prime-power sums with phase $\exp(2\pi i\alpha n^\beta)$ on intervals of length $K^\delta$. For $0<\beta<13/20$, we obtain a power saving whenever $\max\{17/30,1-\beta\}<\delta<1$; in particular the square-root phase admits every $\delta>17/30$. A revised transform estimate retains both the stationary spread and the Fourier width $K/L$. For example, the bound at $\beta=1/2$, $\delta=3/5$ is $K^{15/26+\varepsilon}$, uniformly for frequencies $m\alpha$ with $1\le m\le K^{1/10}$. We also retain power savings for certain self-dual $GL(2)$ twists and self-dual $GL(3)$ cusp forms, without assuming Ramanujan. All three applications have explicit frequency dependence.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-14: 1 request, 6,616 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.34 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We deduce zero-density estimates in short vertical strips from short-interval moments of globally tempered automorphic $L$-functions, by adapting the large-value estimate of Guth and Maynard and the Type II zero argument of Maynard and Pratt to the framework of Ye and Zhang. For the Riemann zeta function, we obtain $N_\zeta(\sigma;T,T+T^a)\ll T^{15a(1-\sigma)/(3+5\sigma)+\varepsilon}$ for $2/3\le a\le1$ and $7/10\le\sigma\le4/5$. This improves Ye--Zhang's exponent for $7/10<\sigma<4/5$ and localizes Guth--Maynard's density bound to short intervals of ordinates. Through the explicit formula, we apply these estimates to smoothed prime-power sums with phase $\exp(2\pi i\alpha n^\beta)$, for fixed $\alpha\ne0$ and $0<\beta<1$. We obtain explicit power savings over the interval length $K^\delta$ for zeta and, using the earlier cusp-form density bounds, for certain real primitive character twists of self-dual $GL(2)$ cusp forms and self-dual $GL(3)$ Maass cusp forms. The smallest interval thresholds in these three applications are $2/3$, $3/4$, and $5/6$, respectively. We give explicit frequency dependence when $\alpha$ is replaced by $m\alpha$, uniformly for $1\le m\le K^R$ with any fixed $R>0$. The cusp-form applications do not require the Ramanujan conjecture.

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	We deduce zero-density estimates in short vertical strips from short-interval moments of globally tempered automorphic $L$-functions, by adapting the large-value estimate of Guth and Maynard and the Type II zero argument of Maynard and Pratt to the framework of Ye and Zhang.	contribution 0.59	0.94	0.33	asserted 1.00
s1	For the Riemann zeta function, we obtain $N_\zeta(\sigma;T,T+T^a)\ll T^{15a(1-\sigma)/(3+5\sigma)+\varepsilon}$ for $2/3\le a\le1$ and $7/10\le\sigma\le4/5$.	result 1.00	0.94	0.38	asserted 1.00
s2	This improves Ye--Zhang's exponent for $7/10<\sigma<4/5$ and localizes Guth--Maynard's density bound to short intervals of ordinates.	interpretation 0.86	0.90	0.30	asserted 1.00
s3	Through the explicit formula, we apply these estimates to smoothed prime-power sums with phase $\exp(2\pi i\alpha n^\beta)$, for fixed $\alpha\ne0$ and $0<\beta<1$.	method 0.72	0.92	0.06	asserted 1.00
s4	We obtain explicit power savings over the interval length $K^\delta$ for zeta and, using the earlier cusp-form density bounds, for certain real primitive character twists of self-dual $GL(2)$ cusp forms and self-dual $GL(3)$ Maass cusp forms.	result 0.96	0.95	0.19	asserted 1.00
s5	The smallest interval thresholds in these three applications are $2/3$, $3/4$, and $5/6$, respectively.	result 1.00	0.90	0.11	asserted 1.00
s6	We give explicit frequency dependence when $\alpha$ is replaced by $m\alpha$, uniformly for $1\le m\le K^R$ with any fixed $R>0$.	result 0.72	0.93	0.04	asserted 1.00
s7	The cusp-form applications do not require the Ramanujan conjecture.	result 0.68	0.63	0.04	asserted 1.00

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

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