On the prime geodesic theorem

Alberto Acosta Reche

REVIEW

On the prime geodesic theorem

Alberto Acosta Reche

Number Theory · Submitted 1 Oct 2026

Pith's reading: This paper claims that the error term in the prime geodesic theorem can be improved unconditionally for the modular group and, more generally, for every congruence subgroup of an indefinite quaternion algebra over $\mathbb{Q}$.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on October 2, 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 →

· 1 major objection · 4 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2610.01998 (version 2)

Pith ID66RNF4ML, Pith's permanent record of this paper and its review

CitationAlberto Acosta Reche. "On the prime geodesic theorem." arXiv:2610.01998 (2026). https://pith.science/paper/66RNF4ML

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Referee report

Significance

If the framework from [Rec26a] and [Rec26b] is correct, the paper delivers a genuine, uniform improvement over [SY13, Theorem 1.1] for every group in the stated class, with a clean closed-form exponent η(θ) and a new zero-density estimate (Proposition 2.1) proved inside the paper from published inputs (Heath-Brown's large sieve, Petrow–Young subconvexity). I verified the dyadic-large-sieve combination in Proposition 2.1 and the final balancing of σ, V, Y; both are internally consistent and give η(1/6) ≈ 0.69359 < 25/36. The central defect is that the main theorem is a reduction to black boxes from unpublished preprints by the same author: the spectral estimate (25), the trace formula (26), the identity (27) and decomposition (30) with the bounds (29) and (31), the asymptotic (35) via [Rec26b, eq. (146)], and the division-algebra reduction [Rec26a, Theorem 1.2]. The advertised claim 'improves unconditionally' is therefore only as strong as those preprints. With that dependence resolved, the paper's own contribution is solid and valuable; as presented, verification of Theorem 1.2 is impossible from this manuscript alone.

Major objections (1)

[§3 (eqs. (25)–(31)) and §4 (eqs. (33), (35))] Theorem 1.2 is reduced to unverified black boxes from the author's own unpublished preprints, and this blocks independent verification of the central claim. The chain is: the spectral estimate S(T,x) ≪ T^{5/4}x^{1/8+ε} in (25) and the trace formula (26) quoted from [Rec26b, Theorem 1.7 and Lemma 5.2]; the identity (27) and the decomposition (30) with the bounds (29) and (31) from [Rec26b, Section 9]; the asymptotic (35) derived from [Rec26b, Equation (146)]; and the division-algebra reduction [Rec26a, Theorem 1.2] used at the start of Section 4. Both Proposition 4.1 and Lemma 3.3 pass through these inputs, and (30) is precisely the point where the subconvexity exponent θ for L(s; ψ_D ψ) is transplanted into the geodesic error term via (29); if (30) isolates the wrong character, or if (29) holds only with a weaker or n-restricted exponent, then (37) and the exponent η(θ) shift or fail. Since [Rec26a] and [Rec26b] are listed as arXiv preprints and are not independently verified, Corollary 1.3's claim of an unconditional improvement over [SY13, Theorem 1.1] should either be stated explicitly as conditional on those preprints, or the paper should supply precise statements (and proofs or proof sketches) of (25), (26), (29), (30), (31) and (35) in an appendix, together with a check that the hypotheses match the usage here (θ admissible, |n| ≥ 3, Re(s) ≥ 1/2, |s−1| ≥ ε).

Minor objections (4)

[§4, final paragraph] The optimization step described as 'a routine calculation' is what produces the numerical exponent η(1/6) ≈ 0.6936. Please display the computation, or at least record the inequalities confirming that for θ = 1/6 the chosen values satisfy σ ∈ [1/2, 1], 1 ≤ V ≤ x, and x^{1/2+ε} ≤ Y ≤ x^{1−ε}, so that a reader can reproduce the claimed gain over 25/36.
[§1.1] The name 'Soundarajan and Young' should read 'Soundararajan and Young'; the phrase 'Lindel¨ of hypothesis' should read 'Lindelöf hypothesis'.
[§1.1, eqs. (13)–(15)] The passage using (15) and the heuristic bound |B_σ| ≤ x^{1/3} is motivational and is not used formally later; it would help to mark it explicitly as a heuristic, since the formal content is Proposition 2.1.
[Corollary 1.3] The comparison line 'η(1/6) ≈ 0.69359 < 0.69 4 = 25/36' has an odd spacing; 25/36 = 0.69444... so it would read more cleanly as 'η(1/6) ≈ 0.69359 < 25/36 = 0.6944...'.

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

The paper

The paper claims

This paper claims that the error term in the prime geodesic theorem can be improved unconditionally for the modular group and, more generally, for every congruence subgroup of an indefinite quaternion algebra over $\mathbb{Q}$.

The reading

This paper claims an unconditional improvement to the error term in the prime geodesic theorem, the asymptotic count of closed geodesics of bounded length on a hyperbolic surface. The main result, Theorem 1.2, states that for any congruence subgroup of an indefinite quaternion algebra over $\mathbb{Q}$, the weighted counting function satisfies $\Psi_\Gamma(x;\varphi)=\langle\varphi,1\rangle x+O_{\varphi,\varepsilon}(x^{\eta(\theta)+\varepsilon})$ with an explicit exponent $\eta(\theta)=\frac12+\frac{\sqrt{81\theta^2+106\theta+33}-5-9\theta}{4}$, for any admissible subconvexity exponent $\theta$. Setting $\theta=1/6$, which the author notes is admissible by Weyl-type bounds for Dirichlet $L$-functions of cube-free conductor, gives $\eta(1/6)\approx 0.69359$, just below the previous best $25/36\approx 0.6944$ of Soundararajan and Young. The point is that the same explicit exponent applies uniformly to every group in that broad class, not only to the modular group.

What carries the argument

The load-bearing object is the exceptional set $B_\sigma=\{n\in(X,X']:L(s,n^2-4)\text{ has a zero in }[\sigma,1]\times[-x^\varepsilon,x^\varepsilon]\}$, together with the new zero-density estimate Proposition 2.1, which bounds $|B_\sigma|$ by $x^{2(1+2\theta)(1-\sigma)+\varepsilon}$ using a subconvexity fourth moment over the exceptional values. Off $B_\sigma$ a standard Lindelof-on-average argument bounds the contour integral $I_V(n^2-4)$ by $x^\varepsilon V^{\sigma-1}$; on $B_\sigma$ the subconvexity bound gives $x^{\theta+\varepsilon}V^{-1/2}$. Balancing the two contributions in the local formula for $\Psi_\Gamma(x+u)-\Psi_\Gamma(x)$, optimized over the parameters $\sigma,V,Y$, produces the exponent $\eta(\theta)$, with its square root coming from a quadratic optimization in $1-\sigma$.

Extended reading notes

Core claim

The paper's central claim is a two-part theorem. Theorem 1.2 asserts that for any congruence subgroup $\Gamma$ of any indefinite quaternion algebra over $\mathbb{Q}$ and any class function $\varphi$ on the finite quotient $\tilde{\Gamma}$, the weighted count satisfies $\Psi_\Gamma(x;\varphi)=\langle\varphi,1\rangle x+O_{\varphi,\varepsilon}(x^{\eta(\theta)+\varepsilon})$ with $\eta(\theta)$ as above, whenever $\theta$ is an admissible subconvexity exponent for quadratic Dirichlet $L$-functions in the level aspect. Corollary 1.3 deduces, for the modular group and uniformly for every group in the class, that $\Psi_\Gamma(x)=x+O_\varepsilon(x^{\eta(1/6)+\varepsilon})$ with $\eta(1/6)\approx 0.69359<25/36$, which the paper describes as an unconditional improvement on the previous record. The proof obtains a new zero-density estimate for quadratic characters (Proposition 2.1) that replaces Heath-Brown's fourth-moment bound $x^{1+\varepsilon}$ by a subconvexity fourth moment $|B_\sigma|x^{4\theta+\varepsilon}$ over an exceptional set $B_\sigma$, and then feeds this through a smoothed trace-formula identity for $\Psi_\Gamma(x+u)-\Psi_\Gamma(x)$.

Load-bearing premise

The argument leans entirely on identities and reductions quoted from the author's own unpublished preprints — the spectral bound for $S(T,x)$, the exact trace formula, the meromorphic continuation and $L$-function decomposition of $Z(s;n)$, and the reduction of division-algebra groups to the split case — and Theorem 1.2 collapses if any of those fails, since none is published or independently verified.

Editorial extensions

If this is right

For the modular group the error term in $\Psi_\Gamma(x)=x+O_\varepsilon(x^{25/36})$ drops to $O_\varepsilon(x^{\eta(1/6)+\varepsilon})$ with $\eta(1/6)\approx 0.69359$, the arithmetic first improvement beyond that bound if the theorem is correct.
The same exponent holds uniformly for every congruence subgroup of every indefinite quaternion algebra over $\mathbb{Q}$, so the improvement is not tied to the modular group alone.
The formula is structural: any future admissible exponent $\theta<1/6$ obtained from stronger subconvexity feeds directly into the same $\eta(\theta)$, so the theorem upgrades automatically as better level-aspect bounds for quadratic Dirichlet $L$-functions become available.
Theorem 1.2 also covers the weighted sums $\Psi_\Gamma(x;\varphi)$ with arbitrary class functions $\varphi$, extending the same error term to Chebotarev-type weightings of geodesic conjugacy classes.

Reading between the lines

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

Editorial inference: even with the hypothetical best subconvexity exponent $\theta=0$ (Lindelof in level aspect), the formula gives $\eta(0)=\tfrac12+(\sqrt{33}-5)/4\approx 0.686$, still above $2/3$; the zero-density mechanism as presented cannot reach the conjectured 'square-root' bound, so a structurally different input would be needed.
Editorial inference: because the proof reduces every division-algebra case to the split case via [Rec26a, Theorem 1.2], the risk is concentrated: if the split preprint's $Z(s;n)$ machinery fails, the improvement is lost for the whole class of groups, not just the modular case.
Editorial inference: since the admissible exponent is known for twists by any fixed character, Proposition 2.1 also applies to $L(s,\chi\psi)$ with fixed $\psi$; a similar short-interval mean-value problem for quadratic characters twisted by a fixed Dirichlet character may admit the same type of improvement.

Original abstract

We improve the error term in the prime geodesic theorem for the modular group. More generally, our result applies to any congruence subgroup of any indefinite quaternion algebra over $\mathbb{Q}$.

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, 2026-10-02: 1 request, 2,203 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.45 · 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.

We improve the error term in the prime geodesic theorem for the modular group. More generally, our result applies to any congruence subgroup of any indefinite quaternion algebra over $\mathbb{Q}$.

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 improve the error term in the prime geodesic theorem for the modular group.	contribution 0.64	0.95	0.87	asserted 1.00
s1	More generally, our result applies to any congruence subgroup of any indefinite quaternion algebra over $\mathbb{Q}$.	result 0.48	0.85	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.