On primeness of the Selberg zeta-function

J\"orn Steuding; Ram\=unas Garunk\v{s}tis

REVIEW 3 major objections 4 minor 14 references

On primeness of the Selberg zeta-function

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Selberg zeta-function of a compact Riemann surface is pseudo-prime and right-prime, forcing any rational outer factor in a decomposition to be a polynomial whose degree divides 2g−2 and the inner factor to be entire.

desk verdict First primeness result for the Selberg zeta-function, likely true but the proof as printed has a repairable gap in the right-primeness step plus several typos; send to a referee but require a fix before acceptance. read the letter →

arxiv 1908.03108 v1 pith:FZLKLIJV submitted 2019-08-08 math.NT

Ramūnas Garunkštis , Jörn Steuding This is my paper · ORCID

classification math.NT MSC 11M36

keywords Selbergzeta-functioncompactRiemannsurfaceprimefunctionpseudo-primeright-primedecompositionvaluedistributiontrivialzeros

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 that the Selberg zeta-function associated with a compact Riemann surface of genus g is pseudo-prime and right-prime in the sense of value-distribution theory: it cannot be written as a nontrivial composition f(h(s)) with meromorphic f and h. In every such decomposition with f rational, f must be a polynomial of degree k dividing 2g−2, and h must be entire. This is the Selberg-zeta analogue of a known primeness result for the Riemann zeta-function, and it implies that the zero structure of Z is irreducible in a strong compositional sense. The proof combines a classical pseudo-primeness criterion with contrasting growth estimates of Z in different sectors, a zero-counting argument on circles around the trivial zeros, and a bound on the number of zeros of exponential-type inner factors.

What carries the argument

The proof is carried by three mechanisms: (i) a classical lemma from complex analysis stating that a finite-order meromorphic function whose two distinct a-value sets have finitely many accumulation lines is pseudo-prime, applied to the zeros and poles of Z; (ii) the contrasting growth behaviour of Z in the right half-plane (Z(s)→1 as σ→∞) and in a sector containing the negative real axis (Z(s)→∞), which forces any polynomial inner factor h to be linear by comparing preimages of a half-line; and (iii) a zero-counting and Rouché argument on the circles |s+n|=1/2 around the trivial zeros, using the fact that Z(s)→∞ on these circles, to show the degree k of a polynomial outer factor must divide 2g−2. Lemma 7, which supplies the growth on those circles, is reduced via the functional equation to a positive lower bound on t + (1/π) log|1−$e^{{2πis}}$| for |s|=1/2, proven by a four-case split.

What would settle it

A direct counterexample would be an explicit decomposition Z(s)=f(h(s)) with rational f of degree not dividing 2g−2 for some compact Riemann surface (e.g., a cubic outer factor for genus 2); short of that, evaluating the Lemma 7 bound at |s|=1/2, arg s=π/3 (where the paper's value 0.41 is wrong, the true value being about 0.365) and finding a non-positive value there would break Lemma 7 and with it the zero-counting proof of the divisibility claim.

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Extended reading notes

Core claim

The central discovery is that the Selberg zeta-function Z(s) of a compact Riemann surface X of genus g is prime in the decomposition sense: any representation Z(s)=f(h(s)) with f and h meromorphic forces one component to be trivial. Specifically, Z is pseudo-prime (every decomposition has f rational or h a polynomial) and right-prime (h is linear whenever f is transcendental); and if f is rational then f is a polynomial of degree k dividing 2g−2 and h is entire. The argument shows that the only possible nontrivial inner factors would be exponentials of linear functions, which are ruled out by the quadratic growth of the zero-counting function of Z, and that polynomial outer factors of degree k would force the zero multiplicities (2g−2)(2n+1) at the trivial zeros s=−n to be divisible by k for all large n, which is possible only if k divides 2g−2.

Load-bearing premise

The proof hangs on Lemma 7's claim that Z(s) becomes unbounded on the circles |s+n|=1/2 as n→−∞, and that claim rests on a numerical lower-bound check for t + (1/π) log|1−$e^{{2πis}}$| on |s|=1/2 that is computed incorrectly in one of the four angular cases, leaving a gap that must be repaired for the proof to be complete.

Editorial extensions

If this is right

The Selberg zeta-function of a compact Riemann surface is pseudo-prime: in every decomposition, either the outer map is rational or the inner map is a polynomial.
Any transcendental outer map forces the inner map to be linear, so the only possible nontrivial decompositions have a polynomial outer map of degree dividing 2g−2.
If two compact Riemann surfaces have Selberg zeta-functions whose nontrivial zeros are connected by an entire map h, then h is the identity and the zero sets coincide.
The value distribution of Z is prime in the sense that its zero sets and preimages cannot be written as unions of level sets of a simpler function, restricting attempts to model the nontrivial zeros by a composition.
The paper's concluding remark suggests the stronger and still open possibility that no nontrivial decomposition exists at all.

Reading between the lines

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

The same growth-dichotomy plus zero-counting strategy may apply to other zeta-functions with an Euler product and a functional equation, such as Selberg zeta-functions for finite-volume non-compact surfaces, provided a suitable analogue of Lemma 7 holds.
In the hypothetical quadratic decomposition, the equality N(d,Z)=N(e,g)∪N(f,g) together with the clustering of a-points suggests that any inner function g would itself need to satisfy a functional equation of Selberg type, which would likely rule out such decompositions entirely.
The numerical error found in the bound of Lemma 7, while repairable, indicates that the proof is delicate at that point; a rigorous re-proof of Lemma 7 is needed before the theorem can be considered fully established.
Corollary 2 shows that the zero set of Z is a complete invariant under entire maps; a natural extension would be to ask whether meromorphic maps (rather than entire) also force h to be the identity.

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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

Here's my read. The genuinely new thing is Theorem 1: the Selberg zeta-function of a compact surface is pseudo-prime and right-prime, and in the rational-component case the polynomial degree divides 2g−2. That's a real addition to the short list of zeta-functions with primeness properties, and the degree-divisibility statement is new. The proof strategy is sensible: pseudo-primeness comes from Liao–Yang's lemma, right-primeness from the contrast between Z→1 in the right half-plane and Z→∞ in the excluded sectors, and Lemma 8 uses Rouché on the trivial-zero circles to force the divisibility. The paper is concise and mostly readable.

But the right-primeness step as written does not work. In Proposition 5 the half-line ℓ is defined as {arg s = π/2 − arg a_d} but the preimage rays L_j are computed as if ℓ had angle π/2. The correct preimage angles of the stated ℓ would be (π/2 − 2 arg a_d + 2jπ)/d. More importantly, the assertion 'arg L_j ≠ π' is false. Take d = 2 and h(s) = i s^2: the preimage of the positive imaginary axis contains the negative real axis, where Z has trivial zeros and certainly does not tend to ∞. The contradiction in (7) then isn't available for that preimage. This is load-bearing for the proof of right-primeness, though I think it's repairable: a generic rotation of the half-line avoids the excluded sector around the negative real axis. The intended dichotomy is right; the execution is missing a step.

Smaller issues: Lemma 7, case 3, writes '√3/4 + log(...)' where the 1/π in front of the log is dropped; with the factor restored the estimate is true, so this is a typo, not a fatal flaw. The reference to 'Lemma 11' in Proposition 5 should be Lemma 4. Corollary 2's 'two accumulation lines, namely arg s = π/2 and 3π/4' looks like a typo for 3π/2, and the application of Lemma 3 with a_2 = ∞ to a function with no poles is underexplained. None of these undermine the central idea.

The self-citations to [4] and [5] are fine; those are independent growth results, not circular.

My verdict: the theorem is probably true and worth a serious referee, but the printed proof needs repair. I'd send it to review and tell the authors to fix the preimage computation in Proposition 5 before acceptance. It's the kind of paper a value-distribution person will want to know about.

Referee Report

3 major / 4 minor

Summary. The paper studies the Selberg zeta-function Z(s) attached to a compact Riemann surface of genus g≥2 in the sense of decomposition (factorization) of meromorphic functions. Theorem 1 claims that Z is pseudo-prime and right-prime, and that if Z(s)=f(h(s)) with f rational and h meromorphic, then f is a polynomial whose degree divides 2g−2 and h is entire. Corollary 2 claims a rigidity property for entire maps sending the nontrivial zero set of one Selberg zeta-function to that of another. The proofs combine a criterion of Liao and Yang on accumulation lines of value sets, growth estimates for Z and the functional-equation factor X(s), and zero-counting on circles centered at negative integers.

Significance. If the main theorem is correct, it gives a strong rigidity statement: every nontrivial decomposition of the Selberg zeta-function has a very restricted outer or inner component, and in the rational outer case the degree is constrained by the genus through 2g−2. The approach is structurally attractive because it avoids special-function identities and instead uses sectorial growth contrasts plus Rouché-type counting. The paper also has a clear falsifiable consequence, the divisibility constraint on the degree of a polynomial outer component. The main ideas are sound and the argument is largely self-contained modulo the authors' earlier estimates, but the proof of right-primeness contains a concrete geometric error that must be repaired before the central claim is fully established.

major comments (3)

[§2, Proposition 5] The preimage-ray computation in the proof of right-primeness is incorrect. For h(s)=a_d s^d+..., the half-line ℓ={arg s=π/2−arg a_d} has preimage rays asymptotic to L_j with angles α_j=(π/2−2 arg a_d+2πj)/d, not (π/2−arg a_d+2πj)/d as stated. Consequently the assertion 'arg L_j ≠ π' is false in general. For d=2 and arg a_d=π/4, the correct preimage rays of ℓ are the positive and negative real axes; the negative real axis is not contained in the sector A, and along that ray Z does not tend to ∞ (it vanishes at every positive integer, while the growth statements in (5) and (6) do not apply). Thus the two limits in (7) cannot both be obtained for this preimage, and the contradiction proving d=1 is not established. Since d=2 is the central case for a quadratic inner factor, this is a load-bearing gap. The gap is repairable by choosing the half-line ℓ with a generic angle so that none of its preimage rays lands on the negative real axis and at least one lies eventually in σ>2 and one in A, but that additional step is absent.
[§2, Lemma 7, case 3] The numerical estimate in case 3 is false as printed: the inequality √3/4 + log(1−e^{−2π√3/4}) ≥ 0.41 is incorrect, since the left-hand side is approximately 0.365. The lemma itself survives because the weaker positive lower bound 0.36 still exceeds the stated δ=0.007. Nevertheless, this is a computational premise in a lemma that is used both in Lemma 8 and in the right-primeness growth argument, so the incorrect estimate should be corrected or replaced by a valid bound.
[§2, proof of Corollary 2] The proof of Corollary 2 is incomplete as written. The set E is said to have accumulation lines arg s=π/2 and 3π/4, but the zeros of the Selberg zeta-function lie essentially on the line σ=1/2, whose accumulation angles are π/2 and 3π/2 (mod 2π), not 3π/4. More importantly, the final step 'Therefore g(s)=bs and b=1' is not justified: after Lemma 9 gives deg g≤2, one must prove that a polynomial of degree at most two that maps all but finitely many points of the line σ=1/2 into the same line must be the identity. This is plausible, but the argument is missing and the notation is confusing. Since the corollary is a stated result, this gap should be addressed.

minor comments (4)

[§2, Proposition 5] The proof refers to 'Lemma 11' when citing the asymptotic for the factor X(s); this should be Lemma 4.
[§2, Lemma 6] The sentence 'Then p is a linear function since F is a right-prime function' is too terse. As written, right-primeness does not apply directly to the displayed decomposition F=f(h) with rational f and transcendental h. The intended argument must re-compose as F(z)=G(p(z)) with G(z)=f(e^z−w) and then apply right-primeness to that decomposition; this step should be stated explicitly.
[§2, Lemma 7, case 4] The displayed equality in case 4 appears to be a typographical repetition of the same expression on both sides; the intended lower bound should be written as an inequality.
[§2, proof of Corollary 2] The angle '3π/4' should presumably be '3π/2', and the reuse of the symbol b for both the Hadamard constant and the coefficient of g is confusing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Selberg zeta-function primeness proof relies on independent growth lemmas and standard factorization criteria; the noted gaps are computational or citation gaps, not circular reasoning.

full rationale

The derivation of Theorem 1 is not circular. The pseudo-primeness conclusion is obtained by applying Lemma 3 (a standard criterion from Chuang et al.) to the Selberg zeta-function; the required finite accumulation-line property of zeros follows from the known zero distribution quoted from Hejhal, not from the decomposition under study. The right-primeness conclusion is derived from the asymptotic (5), the functional equation (3), and the growth estimate for X(s) (Lemma 4, cited as Lemma 1 of the authors' [5]); that cited lemma is a parameter-free asymptotic formula proved independently of factorization and does not assume the primeness conclusion. Lemma 8's divisibility k divides 2g-2 is obtained by comparing zero counts of Z(s) and a_k h(s)^k on the circles |s+n|=1/2 via Lemma 7 and Rouche's theorem; the divisibility is a consequence, not a hypothesis. The Gross/Chuang-Yang definitions are standard terminology, no fitted parameter is renamed a prediction, and no uniqueness theorem from the authors' prior work is imported to forbid alternatives. There are genuine rigor gaps, but they are correctness gaps, not circularity: Proposition 5 invokes a nonexistent 'Lemma 11' after the functional equation (3), the preimage-ray angle computation preceding (7) is wrong as written ('arg L_j != pi' fails for d=2 and arg a_d=pi/4), and Lemma 7 case 3 states 'sqrt(3)/4 + log(1 - e^{-2pi sqrt(3)/4}) >= 0.41', whereas the left side is about 0.365. Each of these needs repair or a supplementary argument, but none makes a theorem equivalent to its own inputs, so they do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper contributes no new free parameters or entities; it is a pure proof that imports standard Selberg zeta-function theory and factorization-theoretic lemmas from the literature.

assumptions (4)

domain assumption Selberg zeta-function properties from Hejhal: Z is entire of order 2, has trivial zeros at negative integers with multiplicity (2g-2)(2n+1), satisfies functional equation (3), and has asymptotic (5) Z(s)=1+O(N(P00)^{-sigma}).
Invoked in Section 1 and in Proposition 5 and Lemma 8; treated as established from the Selberg trace formula literature, not proved in the paper.
domain assumption Lemma 3 (Chuang/Yang): a finite-order meromorphic function with finitely many accumulation lines of the sets {f=a1} and {f=a2} is pseudo-prime.
Used as a black box from [1, p.141] to establish pseudo-primeness of Z in Proposition 5; no proof is given in the paper.
domain assumption Lemma 4 (Garunkstis-Simenas): asymptotic for X(s) in the functional equation.
Used in Proposition 5 to infer growth of Z in the left half-plane; cited from [5], with proof not included.
domain assumption Lemma 9 (Edrei): if an entire function has roots of f(z)=a_j on a straight line for an unbounded sequence a_j, then f is a polynomial of degree at most 2.
Used in Corollary 2 to constrain the polynomial g; cited from [3].

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Pith. "Pith review of On primeness of the Selberg zeta-function." pith.science (2026). https://pith.science/paper/FZLKLIJV

@misc{pith2026190803108,
  author       = {Pith},
  title        = {Pith review of: On primeness of the Selberg zeta-function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FZLKLIJV}},
  note         = {Machine review of arXiv:1908.03108}
}

read the original abstract

In this note we prove that the Selberg zeta-function associated to a compact Riemann surface is pseudo-prime and right-prime in the sense of a decomposition.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [4]

Garunk ˇstis and A

R. Garunk ˇstis and A. Grigutis , The size of the Selberg zeta-function at places sym- metric with respect to the line Re(s) = 1 /2, Results Math. 70 (2016), 271–281

work page 2016

[2] [5]

Garunkˇstis and R

R. Garunkˇstis and R. ˇSim ˙enas, The a-values of the Selberg zeta-function, Lith. Math. J. 52(2) (2012), 145–154

work page 2012

[3] [1]

Chitai Chuang et al , Several Topics in Theory of One Complex Variable , (Science Press, 1995) (Chinese)

work page 1995

[4] [2]

Chi-Tai Chuang and Chung-Chun Yang , Fix-points and factorization of meromorphic functions, World Scientiﬁc Publishing Co., Inc., Teaneck, NJ, (1995)

work page 1995

[5] [3]

Edrei , Meromorphic functions with three radially distributed val ues, Amer

A. Edrei , Meromorphic functions with three radially distributed val ues, Amer. Math. Sot. Trans. 78 (1955), 276–293

work page 1955

[6] [6]

Garunkˇstis, R

R. Garunkˇstis, R. ˇSim ˙enas, and J. Steuding , The a-points of the Selberg zeta-function are uniformly distributed modulo one , Illinois J. Math. 58 (2014), 207–218

work page 2014

[7] [7]

Gross , On factorization of meromorphic functions , Trans

F. Gross , On factorization of meromorphic functions , Trans. Amer. Math. Soc. 131 (1968), 215–222

work page 1968

[8] [8]

Hejhal, The Selberg trace formula for P SL(2,R)

D.A. Hejhal, The Selberg trace formula for P SL(2,R). Vol. 1 , Lecture Notes in Mathe- matics, 548, Springer-Verlag, 1976

work page 1976