Erd\H{o}s-Kac type theorem for the number of scattering geodesics on modular surface

Punya Plaban Satpathy; Sudhir Pujahari

REVIEW 2 major objections 6 minor 8 references

Erd\H{o}s-Kac type theorem for the number of scattering geodesics on modular surface

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that the number of distinct prime factors of the scattering-geodesic count n_q follows a Gaussian distribution, exactly as in the classical Erdős–Kac theorem.

desk verdict The arithmetic core is sound and the reduction is elegant, but the geometric bridge to scattering geodesics is imported from an unpublished preprint. read the letter →

arxiv 2506.07474 v1 pith:W2RXPEHP submitted 2025-06-09 math.NT math.DG

Sudhir Pujahari , Punya Plaban Satpathy This is my paper

classification math.NTmath.DG MSC 11N4511N1311F72

keywords Erdős–KactheoremscatteringgeodesicsmodularsurfaceEulertotientfunctionL-functionsojourntimenormaldistributionarithmeticfunctions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a hyperbolic-geometry analogue of the classical Erdős–Kac theorem. Let $n_q = (\phi(q)+s_q)/2$ be the number of scattering geodesics on the modular surface whose sojourn time is $2\log(qT_0)$; the theorem states that, as $q$ ranges up to $x$, the number of distinct prime factors $\omega(n_q)$ is normally distributed with mean $\frac{1}{2}(\log\log x)^2$ and variance $\frac{1}{3}(\log\log x)^3$. The result matters because it transplants a central law of probabilistic number theory into the counting of geodesic trajectories, showing that an arithmetic statistic of the geodesic counts follows the same bell curve as prime factors of ordinary integers. The proof works by transferring an existing Erdős–Kac theorem for Euler's totient function to $n_q$, using a correspondence between scattering geodesics and rational numbers established in [8] and a sharp estimate on how often the correction term $s_q$ is nonzero.

What carries the argument

The load-bearing objects are the arithmetic function $n_q = (\phi(q)+s_q)/2$ and the exceptional set it defines. Here $\phi$ is Euler's totient and $s_q$ counts solutions of $p^2 \equiv -1 \pmod q$. The geometric input is a bijection, established in [8], between scattering geodesics $\mathcal{S}$ and a set of rationals $\mathcal{G} = \bigsqcup_q \mathcal{G}_q$, under which the $q$-th block $\mathcal{G}_q$ has exactly $n_q$ elements and corresponds to geodesics with common sojourn time $2\log(qT_0)$. The arithmetic input is the Dirichlet series $F(s) = (1+2^{-s})\sum_{q\in \mathcal{O}} q^{-s}$, whose square equals $(1+2^{-s})L(s,\chi)\zeta(s)G(s)/\zeta(2s)$; a Tauberian argument turns the simple pole of $F(s)^2$ at $s=1$ into $A(x) = |\{q \le x : s_q \ne 0\}| \sim \alpha x/\sqrt{\log x}$. This sparsity of the correction term is what lets the proof compare $\omega(n_q)$ with $\omega(\phi(q))$ and import the Erdős–Pomerance normal law.

What would settle it

Compute, for a single large $x$ (say $10^9$), the empirical distribution of $\frac{\omega(n_q) - \frac{1}{2}(\log\log x)^2}{\frac{1}{\sqrt{3}}(\log\log x)^{3/2}}$ over $1 \le q \le x$ and compare its mean and variance with $0$ and $1$; if the variance does not grow like $\frac{1}{3}(\log\log x)^3$, the normalization is wrong. Equally decisive: check numerically whether the proportion $A(x)/x$ of $q \le x$ with $s_q \ne 0$ decays like $\alpha/\sqrt{\log x}$; if it instead stays bounded below by a positive constant, the exceptional set is not $o(x)$ and the transfer from $\omega(\phi(q))$ to $\omega(n_q)$ collapses.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: for every real $a$, the proportion of $q \le x$ for which $\frac{\omega(n_q) - \frac{1}{2}(\log\log x)^2}{\frac{1}{\sqrt{3}}(\log\log x)^{3/2}} \le a$ converges to the standard normal cumulative distribution $\Phi(a)$. The discovery is that the arithmetic function $n_q = (\phi(q)+s_q)/2$, which counts scattering geodesics of a fixed sojourn time, inherits the Gaussian fluctuation law of the Erdős–Pomerance theorem for $\omega(\phi(q))$. The mechanism is that $n_q$ and $\phi(q)$ have almost the same prime divisors: outside a set of $o(x)$ exceptional $q$, one has $|\omega(n_q) - \omega(\phi(q))| \le 1$, because $s_q \ne 0$ only when all prime factors of $q$ (or of $q/2$) are congruent to $1 \bmod 4$, and such $q$ number only $\sim \alpha x/\sqrt{\log x}$.

Load-bearing premise

The argument takes as a black box the previously established correspondence that scattering geodesics split into blocks $\mathcal{S}_q$ of size $n_q = (\phi(q)+s_q)/2$ with common sojourn time $2\log(qT_0)$; if that geometric counting rule were wrong, the main theorem would be a statement about the arithmetic function $n_q$ rather than about geodesics.

Editorial extensions

If this is right

The normal order of $\omega(n_q)$ is $\frac{1}{2}(\log\log q)^2$, so almost every $n_q$ has about $\frac{1}{2}(\log\log q)^2$ distinct prime factors.
The limiting law matches the Erdős–Pomerance distribution for $\omega(\phi(q))$ exactly; the correction $s_q$ does not alter the Gaussian limit.
For all but $o(x)$ values of $q \le x$, one has $|\omega(n_q) - \omega(\phi(q))| \le 1$, so the two arithmetic functions are statistically indistinguishable for this purpose.
The exceptional set $\{q \le x : s_q \ne 0\}$ has size $\sim \alpha x/\sqrt{\log x}$, which is $o(x)$ yet large enough that the correction is present at a nontrivial scale.

Reading between the lines

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

The transfer mechanism suggests a broader principle: any geometric counting function expressible as $(\phi(q) \pm s_q)/2$ with $s_q$ rarely nonzero should inherit every Erdős–Kac-type theorem proved for $\omega(\phi(q))$; this could be tested on congruence subgroups of $\mathrm{PSL}(2,\mathbb{Z})$, where the totient-like counts are governed by other quadratic congruences.
A quantitative version of the theorem is within reach: combining the present exceptional-set estimate with a Berry–Esseen bound for $\omega(\phi(q))$ would yield an effective rate of convergence for the geodesic counts, though the paper does not carry this out.
Because $s_q \ne 0$ only on a sparse set, conditioning on the exceptional $q$ with $p^2 \equiv -1 \pmod q$ might reveal a different distributional law with its own scaling, effectively a mixture-model limit; the paper leaves this unexplored.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a clean, modest paper. If you accept the bridge from [8], the main theorem follows from Erdős-Pomerance plus a density-zero estimate for the exceptional set. The arithmetic proof itself is correct in spirit, with a couple of gaps that are easy to fill.

What's actually new: the statement that ω(n_q) obeys a Gaussian law with mean (1/2)(log log x)^2 and variance (1/3)(log log x)^3, where n_q=(φ(q)+s_q)/2. The reader is right that on a density-one set n_q=φ(q)/2, so the theorem reduces to EP. The genuinely new input is Proposition 2.2: the count of q with s_q≠0 is O(x/√log x), which is o(x). That estimate is standard Dirichlet-series/Tauberian work, and the computation of the constant α looks right. The paper deserves credit for isolating exactly the exceptional set and dealing with it.

Soft spots: three, in increasing order. (1) The shrinking-window step: the paper claims |E1|+|E2|=o(x) directly from Proposition 2.4, but Proposition 2.4 only gives distribution convergence at a fixed a. The argument needs the continuity of Φ plus a standard two-epsilon squeeze. It's true, but the paper should write it out. (2) Kable's Tauberian theorem is invoked without checking its hypotheses (non-negativity of a_q and the precise singularity conditions). That's a minor omission; the hypotheses are plausibly met. (3) The phrase 'clear from heuristic arguments' before the computation of F(s)^2 is sloppy; the subsequent computation is rigorous enough, but the prose should say so.

The bigger caveat, which the paper is honest about: Theorem 1.1, the bijection between scattering geodesics and the sets S_q with sojourn time 2 log(qT_0), is quoted from the authors' preprint [8]. Neither the proof nor any independent check appears here. If that correspondence were wrong, Theorem 1.2 would just be a statement about an arithmetic function. I don't think that's a reason to reject, especially since [8] is a companion paper and the arithmetic statement stands on its own, but a referee should ask for the bridge to be either proved or clearly referenced with a rigorous pointer.

Also: the GitHub URL for the histogram is missing from the text. That's easy to fix.

Bottom line: this is a solid, incremental contribution. It will appeal to people working on arithmetic statistics on hyperbolic surfaces and on Erdős-Kac-type theorems. It deserves peer review. I would accept it with minor revision, asking for the continuity argument to be spelled out, the Tauberian hypotheses verified, and the code URL provided.

Referee Report

2 major / 6 minor

Summary. The paper proves an Erdős–Kac type theorem for the arithmetic function n_q=(φ(q)+s_q)/2, where s_q counts the solutions to x^2≡−1 (mod q). Theorem 1.2 states that for each real a the frequency of q≤x for which (ω(n_q)−(1/2)(log log x)^2)/((1/√3)(log log x)^{3/2})≤a tends to Φ(a). The proof compares ω(n_q) with ω(φ(q)), whose limiting distribution is the Erdős–Pomerance theorem, and uses a Tauberian estimate (Proposition 2.2) for the number of q with s_q≠0 to show that the difference is negligible. The geometric interpretation is supplied by Theorem 1.1, quoted from the authors' preprint [8], according to which the scattering geodesics on the modular surface decompose into sets S_q of cardinality n_q with common sojourn time 2 log(qT0).

Significance. The main idea is transparent and the comparison argument in Section 3 is correct conditional on Proposition 2.2. If the two external inputs were fully established, the result would be a novel and attractive bridge between arithmetic statistics and the geometry of scattering geodesics, and it would fit the journal's scope well. The paper does not ship machine-checked proofs, and the advertised computational evidence is not reproducible because the GitHub URL is omitted. The major barriers are the unverified Tauberian hypotheses in Proposition 2.2 and the complete dependence of the geometric interpretation on an unpublished preprint.

major comments (2)

[§2, proof of Proposition 2.2] The proof of the asymptotic A(x)∼αx/√log x is not complete. The displayed computation shows only that (F(s))^2 has a simple pole at s=1 with the stated residue, but the intended conclusion concerns F(s) itself. The authors invoke Kable's Tauberian theorem [6, Theorem 2] without stating its hypotheses or checking that F(s) satisfies them; in particular, the sentence 'It is also clear from heuristic arguments that F(s) has singular behaviour at s=1' is not a proof. This is load-bearing because Proposition 2.3 and hence the o(x) estimate in Case (I) of Section 3 depend on it. The authors should either verify the conditions of [6, Theorem 2] explicitly or replace the appeal by a self-contained Selberg–Delange argument.
[§1, Theorem 1.1 and reliance on [8]] The geometric statement of the main theorem rests entirely on Theorem 1.1, which is quoted without proof from the authors' preprint [8, Theorem 1.5]. The assertion that the scattering geodesics decompose into sets S_q of cardinality n_q=(φ(q)+s_q)/2, each with common sojourn time 2 log(qT0), is exactly the bridge that makes Theorem 1.2 a theorem about scattering geodesics rather than about the arithmetic function n_q. As written, the present paper cannot be independently verified on this point. The authors should include a proof of Theorem 1.1 (or of the underlying bijection and sojourn-time computation), or make the conditional dependence on [8] explicit and provide the preprint's proof in a form accessible to the referee. In addition, Proposition 2.1 is also quoted from [8] rather than proved here.

minor comments (6)

[§1, end of Introduction] The sentence 'The Python script used to generate the plot shown in Figure 2 is publicly available at GitHub repository' omits the URL; without the repository identifier the numerical evidence cannot be checked.
[§3, Case (II)] The conclusion 'Using Proposition 2.4, we have |E1|+|E2|=o(x)' is correct but deserves one line of justification: since a±1/g(x)→a and Φ is continuous, the proportion of q in that shrinking interval tends to 0.
[§2, proof of Proposition 2.2] Even after the Tauberian hypotheses are verified, the phrase 'clear from heuristic arguments' should be removed or replaced by a rigorous statement, as it is inconsistent with the formal proof style.
[Throughout] The paper should indicate in the introduction which theorems are proved in this paper and which are quoted from [8], so that the reader can see the exact logical dependencies.
[Title page] The Mathematics Subject Classification is listed as 2010; it should be updated to the current 2020 classification.
[Abstract] The phrase 'with a common sojourn time' could be misread; it should be clarified that the sojourn time 2 log(qT0) is common to the n_q geodesics inside each S_q, while it varies with q.

Circularity Check

2 steps flagged · score 4.0 of 10

The arithmetic proof is self-contained and not circular, but the geometric interpretation relies on a load-bearing self-citation ([8]) for the bijection and sojourn-time formula.

self citation load bearing [Section 1.2, Theorem 1.1, recalled from [8, Theorem 1.5]]
"Recently, we [8] have obtained a Prime number theorem kind result for scattering geodesics. For the reader's convenience, we recall some key details from [8] ... Theorem 1.1. Let S be the set of scattering geodesics in M. For each integer q ≥ 1, there exists a subset S_q ⊂ S consisting of n_q := (ϕ(q)+s_q)/2 distinct scattering geodesics, each with common sojourn time 2 log(qT0)."

The paper's advertised object—the number of scattering geodesics with a common sojourn time—is connected to the arithmetic function n_q only through Theorem 1.1, which is quoted from the authors' own preprint [8] without proof. If that bijection or the sojourn-time formula were wrong, Theorem 1.2 would still be a valid Erdős–Kac theorem for the arithmetic function n_q, but not for scattering geodesics. Thus the geometric interpretation of the main result is load-bearing on an unverified self-citation. The distribution result itself is not assumed, however, so the arithmetic derivation is not circular.
self citation load bearing [Section 2, Proposition 2.1, recalled from [8, Theorem 2.2]]
"We recall the following result, which was established in [8, Theorem 2.2]; Proposition 2.1. Let s_q be the arithmetic function defined in Theorem 1.1. Then s_q ≠ 0 if and only if q ∈ O or q/2 ∈ O ..."

The support characterization of s_q is quoted from the authors' prior work and is used to derive Proposition 2.2, the asymptotic A(x) ∼ αx/√log x. Proposition 2.2 is then essential in Proposition 2.3 to show that the exceptional set E(x) has size o(x). Without [8, Theorem 2.2], the proof of the main theorem collapses. This is a second load-bearing self-citation, though again it is a concrete arithmetic fact rather than an assumption of the target Gaussian law.

full rationale

The proof of Theorem 1.2 is not circular in its core arithmetic content. The target distribution of ω(n_q) is reduced to the external Erdős–Pomerance theorem (Proposition 2.4) via a comparison of the normalized indicator sums. Proposition 2.3, which controls the difference between ω(n_q) and ω(φ(q)), uses the support characterization of s_q from [8] and a Tauberian estimate to show that the exceptional set is negligible; the target Gaussian law is never assumed. The central claim is therefore an independent arithmetic theorem about n_q = (φ(q)+s_q)/2. The circularity-adjacent concern is semantic: the paper's stated subject, scattering geodesics, enters only through the authors' own unproved preprint [8], specifically the bijection S ↔ G and the sojourn-time formula 2 log(qT0). This makes the advertised geometric conclusion reliant on self-citation, but it does not force the distribution result, which retains independent content. The missing GitHub URL for the figure is a reproducibility gap, not a circularity. Overall, the derivation is largely self-contained, with load-bearing self-citations that warrant a modest score rather than a charge of full circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the geometric-to-arithmetic correspondence from the authors' earlier work [8] and on two external analytic theorems (Erdős-Pomerance, Kable). No free parameters are fitted; all constants are derived or quoted.

assumptions (5)

domain assumption The scattering geodesics S decompose into subsets S_q with |S_q| = n_q = (ϕ(q)+s_q)/2, each with common sojourn time 2 log(qT0).
Quoted as Theorem 1.1 from [8, Theorem 1.5]; not proved in this paper. This is the bridge between the geometric object and the arithmetic function n_q.
domain assumption s_q ≠ 0 if and only if q ∈ O or q/2 ∈ O, where O consists of integers all of whose prime factors are 1 mod 4.
Quoted from [8, Theorem 2.2] (Proposition 2.1 here); used to compute A(x).
standard math Erdős-Pomerance theorem: ω(ϕ(q)) obeys the stated Gaussian law (Proposition 2.4, [3, Theorem 3.2]).
External theorem used as the benchmark distribution.
standard math Kable's Tauberian theorem ([6, Theorem 2]) converts the singularity of F(s)^2 into the asymptotic A(x) ~ α x / sqrt(log x).
Applied without explicit condition verification.
standard math Standard analytic properties of ζ(s) and L(s,χ) modulo 4.
Used in the expression for F(s)^2.

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Cite this review

Pith. "Pith review of Erd\H{o}s-Kac type theorem for the number of scattering geodesics on modular surface." pith.science (2026). https://pith.science/paper/W2RXPEHP

@misc{pith2026250607474,
  author       = {Pith},
  title        = {Pith review of: Erd\Hos-Kac type theorem for the number of scattering geodesics on modular surface},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W2RXPEHP}},
  note         = {Machine review of arXiv:2506.07474}
}

abstract

In 1917, Hardy and Ramanujan showed that if $\omega(n)$ is the number of distinct prime factors of a randomly chosen positive integer $n,$ then the normal order of $\omega(n)$ is $\log \log \, n.$ This led Erd\H{o}s and Kac to prove their celebrated result showing a Gaussian behaviour for $\omega(n).$ In this article we prove an Erd\H{o}s-Kac kind result for the number of scattering geodesics on the modular surface with a common sojourn time.

Figures

Figures reproduced from arXiv: 2506.07474 by the authors.

**Figure 1.** Fundamental domain for the PSL(2,Z) action on H The object in which we are interested is the modular surface defined as the quotient space M = H/PSL(2, Z) and we assign it with the metric inherited from the upper half-plane. It is a standard result that M is non-compact; however it has a finite area with respect to the measure induced from the hyperbolic metric on H. Consider the natural projection map π : H −→ M. B… view at source ↗

**Figure 2.** Density Histogram of ω(nq) − 1 2 (log log N) 2 √ 1 3 (log log N) 3/2 with 1 ≤ q ≤ N = 107 . The Python script used to generate the plot shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[8]

Prime scattering geodesic theorem

Sudhir Pujahari and Punya Plaban Satpathy,Prime scattering geodesic theorem, 2025, (communicated), arXiv:2505.04973. School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar, An OCC of Homi Bhabha National Institute, P. O. Jatni, Khurda 752050, Odisha, India. Email address : spujahari@niser.ac.in Visitor, School o...

work page Pith review arXiv 2025
[1]

James W Anderson,Hyperbolic geometry, 2 ed., Springer Undergraduate Mathematics Series, Springer, London, England, August 2005

work page 2005
[2]

Erd˝ os and M

P. Erd˝ os and M. Kac,The gaussian law of errors in the theory of additive number theoretic functions, American Journal of Mathematics62 (1940), no. 1, 738–742

work page 1940
[3]

Math.15 (1985), no

Paul Erd˝ os and Carl Pomerance,On the normal number of prime factors of ϕ(n), Rocky Mountain J. Math.15 (1985), no. 2, 343–352

work page 1985
[4]

Victor W Guillemin,Sojourn times and asymptotic properties of the scattering matrix , Publ. Res. Inst. Math. Sci.12 (1976), no. 99, 69–88

work page 1976
[5]

G. H. Hardy and S. Ramanujan,The normal number of prime factors of a number n , Quarterly Journal of Mathematics48 (1917), 76–92

work page 1917
[6]

Kable, A variation of the Ikehara-Delange Tauberian theorem and an application, Comment

Anthony C. Kable, A variation of the Ikehara-Delange Tauberian theorem and an application, Comment. Math. Univ. St. Pauli57 (2008), no. 2, 137–146. MR 2492577

work page 2008
[7]

Z.305 (2023), no

M Ram Murty, V Kumar Murty, and Sudhir Pujahari,An all-purpose Erd ¨os-Kac theorem, Math. Z.305 (2023), no. 3

work page 2023

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [8]

Prime scattering geodesic theorem

Sudhir Pujahari and Punya Plaban Satpathy,Prime scattering geodesic theorem, 2025, (communicated), arXiv:2505.04973. School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar, An OCC of Homi Bhabha National Institute, P. O. Jatni, Khurda 752050, Odisha, India. Email address : spujahari@niser.ac.in Visitor, School o...

work page Pith review arXiv 2025

[2] [1]

James W Anderson,Hyperbolic geometry, 2 ed., Springer Undergraduate Mathematics Series, Springer, London, England, August 2005

work page 2005

[3] [2]

Erd˝ os and M

P. Erd˝ os and M. Kac,The gaussian law of errors in the theory of additive number theoretic functions, American Journal of Mathematics62 (1940), no. 1, 738–742

work page 1940

[4] [3]

Math.15 (1985), no

Paul Erd˝ os and Carl Pomerance,On the normal number of prime factors of ϕ(n), Rocky Mountain J. Math.15 (1985), no. 2, 343–352

work page 1985

[5] [4]

Victor W Guillemin,Sojourn times and asymptotic properties of the scattering matrix , Publ. Res. Inst. Math. Sci.12 (1976), no. 99, 69–88

work page 1976

[6] [5]

G. H. Hardy and S. Ramanujan,The normal number of prime factors of a number n , Quarterly Journal of Mathematics48 (1917), 76–92

work page 1917

[7] [6]

Kable, A variation of the Ikehara-Delange Tauberian theorem and an application, Comment

Anthony C. Kable, A variation of the Ikehara-Delange Tauberian theorem and an application, Comment. Math. Univ. St. Pauli57 (2008), no. 2, 137–146. MR 2492577

work page 2008

[8] [7]

Z.305 (2023), no

M Ram Murty, V Kumar Murty, and Sudhir Pujahari,An all-purpose Erd ¨os-Kac theorem, Math. Z.305 (2023), no. 3

work page 2023