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REVIEW 3 major objections 2 minor 46 references

Power sums and Siegel-type zero-free regions for L-functions

T0 review · 3 major / 2 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper establishes uniform conductor-power Siegel-type zero-free regions for all standard and Rankin–Selberg L-functions, without assuming modularity of tensor products.

desk verdict The power-sum mechanism is genuinely novel and the claimed theorems would be major, but the paper contains a false L-function identity (4.7) that is used in central places, though it looks repairable because only real zeros are involved. read the letter →

arxiv 2608.12257 v1 pith:Q5KTNOLY submitted 2026-08-12 math.NT

Jesse Thorner This is my paper · ORCID
classification math.NT MSC 11M4111F6611M2611R42
keywords zero-freeregionsRankin-SelbergL-functionspowersumsSiegelzerosautomorphicrepresentationsBrauer-SiegeltheoremprimenumbertheoremsTurántheory
topics The Riemann Hypothesis
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 develops a new method for zero-free regions near Re(s)=1 for standard and Rankin–Selberg L-functions attached to cuspidal automorphic representations. The central claim is that for every fixed π′ and every ε>0, there exists an ineffective constant c such that L(s,π×π′) has no zeros and satisfies |L(σ,π×π′)| ≥ c $C_π^{{-ε}}$ in the region σ ≥ 1 − c $C_π^{{-ε}}$; the case π′=1 gives the same for standard L-functions. The method replaces the classical nonnegative-coefficient auxiliary series, whose utility is limited by unproved modularity of Rankin–Selberg products, with two complementary power-sum lower bounds. If correct, these are the first unconditional conductor-power Siegel-type zero-free regions for all GL(n) L-functions, and they imply improved prime number theorems and new Brauer–Siegel-type results.

What carries the argument

The machinery is the pairing of two power-sum inequalities from Section 2. The first, Proposition 2.1, asserts that for any complex numbers z_1,...,z_ν there exists a power ℓ ∈ [K,2K] with |z_1^ℓ+···+z_ν^ℓ| ≥ (|z_1|/50)^ℓ; it is used to extract a single exceptional zero β_ε from the logarithmic derivative G_k of F(z)=L(z,π_ε×π′)L(z,π_ε×~π). The second, Proposition 2.2, is a real-part power-sum bound with nonnegative weights that gives an index j with Re(Σ b_n z_n^j) ≥ b_1|z_1|^j/8; via the zero-repulsion lemma it supplies the factor 1−β_1 that bounds the same G_k from above. Matching the two bounds forces 1−β_1 to be at least a fixed power of $C_π^{{-1}}$, closing the zero-free region.

What would settle it

Because the constant c_25 in Proposition 5.1 is effectively computable, the inequality (5.2) can be tested numerically for a concrete pair (π,π′) with explicitly known Hecke eigenvalues, such as F=Q, π′=1, and π the L-function of an elliptic curve with rational coefficients. Compute both sides at x=(C_π C_{π′})^{c_25} using a rigorous zero-finder to locate the greatest real zero β_1; a violation of (5.2) would identify a defect in the proof, while repeated agreement across several conductors would corroborate the mechanism.

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Extended reading notes

Core claim

The discovery, on the paper's own terms, is that two lower bounds for power sums can serve as the sole engine for zero-free regions: the first power-sum inequality detects the presence of an exceptional zero through the logarithmic derivative of a shifted Rankin–Selberg product, while a real-part power-sum bound, in combination with a zero-repulsion lemma, controls how close any other zero can come to that exceptional zero. The interaction of these two bounds forces the exceptional zero to lie at distance at least a fixed power of the conductor from 1, producing the bound |L(σ,π×π′)| ≥ c $C_π^{{-ε}}$ throughout σ ≥ 1 − c $C_π^{{-ε}}$. Standard analytic properties of Rankin–Selberg L-functions are used only as input, and the classical Siegel–Tatuzawa bound for Dirichlet L-functions is recovered as a special case.

Load-bearing premise

The proof leans on the zero-repulsion estimate for the isobaric object Π = eπ ⊞ π′: every zero of L(s,Π×eΠ) other than the exceptional one is at distance at least a constant multiple of (1−β_1)/log(conductor) from s=1, uniformly in the conductors.

Editorial extensions

If this is right

  • For self-dual π, the prime number theorem error E(x;π) satisfies E(x;π) ≤ c e^{-√log x} as soon as log x ≥ c C_π^ε, matching the quality of classical results for real Dirichlet characters.
  • For Rankin–Selberg prime sums, new ‘highly uniform’ prime number theorems hold, with error term either e^{-√log x} or (log x)^{-1/ε} depending on the self-duality of the factors.
  • The theorem yields Brauer–Siegel-type limits: along any sequence with C_{π_j} → ∞, we have log|L(1,π_j)| / log C_{π_j} → 0 and the analogous statement for L(1,π_j×π′).
  • The result subsumes the twist-aspect Siegel-type bounds and improves the previous conductor-exponent in the fixed-π′ aspect from a fixed power of C_π to an arbitrary ε-power.
  • In the conjectural Bloch–Kato framework for weight −2 motives, the lower bound gives polynomial-in-conductor control of the Shafarevich–Tate group order, paralleling the classical consequence for class numbers and regulators.

Reading between the lines

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

  • If the power-sum mechanism is as robust as the paper suggests, any future improvement in the constants of the underlying power-sum inequalities would transfer directly to wider zero-free regions, effectively decoupling progress on Siegel zeros from progress on functoriality.
  • The same two-inequality architecture might apply to L-functions without a full Rankin–Selberg theory, such as symmetric powers, provided a zero-repulsion statement can be established for them; the paper does not claim this extension.
  • Because the region width is C_π^{-ε}, the proof is uniform in the eigenvalue aspect as well as the conductor aspect; one could test whether the argument adapts to the spectral aspect for Maass forms, where the conductor grows differently.
  • The paper states, with proof deferred to forthcoming work, that nonnegativity-based methods cannot reach the main theorem without an unproven modularity hypothesis; if that meta-claim is correct, the power-sum route is not merely an alternative but the only currently viable path to such unconditional results.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 2 minor

Summary. The paper claims to establish, for every ε > 0, ineffective constants c and c' such that the standard L-function L(s,π) and the Rankin–Selberg L-function L(s,π×π') (with π' fixed) satisfy lower bounds of size (C_π(|t|+3))^{-ε} in a one-sided region σ ≥ 1 - c(C_π(|t|+3))^{-ε}. The method combines two Turán-type power-sum inequalities (Propositions 2.1 and 2.2) with an auxiliary isobaric representation, a Landau–Page-type zero-separation result (Proposition 4.6), and a zero-repulsion estimate (Proposition 5.1) that are intended to force the crucial factor (1−β1). Applications to prime number theorems and Brauer–Siegel-type statements are also presented.

Significance. If the main theorems were correct, they would constitute a major advance: the first unconditional Siegel-type zero-free regions with conductor-power saving for all GL(n) standard and Rankin–Selberg L-functions, subsuming prior results of Brumley and of Harcos–Thorner. The two-pronged power-sum strategy is original, and the paper is clearly organized with careful use of standard analytic number theory tools. However, the proof depends on at least two false L-function identities and one unjustified estimate, so the central claims are not established as written.

major comments (3)
  1. [§4.2, Eq. (4.7)] The identity L(s,eπ×eπ') = L(s,π×π') is false in general. The correct relation is L(s,eπ×eπ') = \overline{L(\bar{s},π×π')}, and for GL(1) Hecke characters χ,ψ the asserted identity would say L(s,\bar{χ}\bar{ψ}) = L(s,χψ), which fails whenever χψ is non-real. This identity is used in the proof of Proposition 4.6 to convert zeros of L(s,π1×π3)L(s,π2×π3) into zeros of L(s,eπ1×eπ3)L(s,eπ2×eπ3), and in the proof of Proposition 5.1 to identify β1 as a double zero of L(s,Π×eΠ). Without (4.7), the four-zero contradiction with Lemma 4.5 and the asserted (1−β1) factor are unsupported.
  2. [§5, proof of Proposition 5.1] The proof asserts that |λ_{eπ}(p)+λ_{π'}(p)|^2 log Np = Λ_{Π×eΠ}(p) by (4.11), but the four Euler factors actually give Λ_{Π×eΠ}(p)/log Np = |λ_π(p)|^2 + |λ_{π'}(p)|^2 + 2Re(λ_π(p)λ_{π'}(p)), whereas the left-hand side equals |λ_π(p)|^2 + |λ_{π'}(p)|^2 + 2Re(\overline{λ_π(p)}λ_{π'}(p)). These expressions differ whenever Im(λ_π(p))Im(λ_{π'}(p)) ≠ 0, so the inequality used to bound the desired sum by S(x;Π) does not follow from (4.11). This is a second load-bearing error in the proof of (5.2).
  3. [§5, proof of Proposition 5.1, contribution of 1−β1] The estimate (x^{−12000β1}−x^{−β1})Γ(−β1) ≪ (1−β1)^{−1}x^{−β1} ≪ 1−β1 is not justified. For β1 close to 1, Γ(−β1) ≍ (1−β1)^{−1}, and the hypotheses x ≥ (CπCπ')^{c25} and (5.1) do not imply x^{−β1} ≤ C(1−β1)^2. If 1−β1 is exceptionally small, this term can be much larger than the claimed O((1−β1)(log x)^3), so the bound (5.2) is not established even if the earlier identity issues were repaired.
minor comments (2)
  1. [§6.1, Eq. (6.1)] The sentence containing 'Otherwise,C pi is bounded' contains a typo: it should read 'C_π'.
  2. [§2.2, Step 2] The quantity N_ε = exp(K/(300ε)) is used before it is formally introduced in Proposition 6.3; flagging it in the strategy section is helpful, but the forward reference is not a substantive issue.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof does not reduce to its inputs; the load-bearing defect is a false identity, not circular dependence.

full rationale

The central derivation does not use Theorem 1.1 or Theorem 1.2 as an input. It starts from independent prior results — Jacquet-Shalika nonvanishing, Brumley's narrow zero-free region (1.6), the Harcos-Thorner twist-aspect theorem (1.7), Moreno's zero-repulsion lemma (Lemma 4.12), Montgomery's power-sum propositions, and Jiang's coefficient bounds (Lemma 4.2) — and then combines the two power-sum lower bounds to force a contradiction with an assumed exceptional zero. The self-citations present, chiefly Harcos-Thorner [12, 13], Lemma 4.5 from [12], and Lemma 4.3 from [13], are parameter-free prior theorems and lemmas whose stated assumptions do not include the target zero-free region, so under the review rules they count as independent evidence rather than circularity. The paper itself flags the relevant limitation in Section 2.2: Bombieri's residue approach cannot be used in this generality without knowing Theorem 1.2 in advance, and the proof therefore switches to Moreno's zero repulsion; this is an explicit acknowledgment of a potential circularity trap, not the commission of one. The genuine vulnerability is non-circular: identity (4.7), L(s,eπ×eπ') = L(s,π×π'), is asserted as "straightforward" but is false for non-real L-functions, such as GL(1) Dirichlet characters, and it is load-bearing in Propositions 4.6 and 5.1. Since the assigned task is circularity rather than general correctness, the false identity does not raise the circularity score; no fitted parameter is renamed as a prediction and no input is equivalent to the output by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central proof is a chain of cited analytic-number-theory results. The paper contributes the power-sum comparison and the contradictions. The main unstated premise that fails is the coefficient identity in Proposition 5.1, which is connected to the false equality (4.7).

assumptions (6)
  • domain assumption Standard analytic properties of GL(n) standard L-functions and Rankin-Selberg L-functions: Euler products, analytic continuation, functional equations, Hadamard factorizations, and conductor bounds (4.8).
    Invoked throughout Section 4 and needed for Lemmas 4.7-4.11.
  • domain assumption The bound |alpha_{j,pi}(p)| <= Np^{theta_n} and Re(mu_{j,pi}(v)) >= -theta_n with theta_n < 1/2 - 1/(n^2+1), from [28,31].
    Used for truncation and error terms, e.g., (4.5) and Proposition 6.3.
  • ad hoc to paper Identity (4.7): L(s,e pi × e pi') = L(s, pi × pi').
    Stated as straightforward but false as an equality of Dirichlet series for non-self-dual representations; only real-zero sets coincide up to conjugation. This feeds into the false prime-coefficient assertion in Proposition 5.1.
  • domain assumption Moreno's zero repulsion theorem (Lemma 4.12), from [30].
    Provides the Deuring-Heilbronn repulsion for the auxiliary L(s,Pi×tilde Pi); the proof relies on Proposition 2.2.
  • domain assumption Harcos-Thorner twist-aspect lower bounds (1.7) from [12,13] and Brumley's narrow zero-free regions (1.6) from [5,26].
    Used in initial reductions in Section 6.1 and Step 1 of Section 2.
  • domain assumption Lemma 4.5 from [12] on zeros of L(s,Pi×tilde Pi) when that function has a pole of order r at s=1.
    Used in Proposition 4.6 to control real zeros near 1.

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Pith. "Pith review of Power sums and Siegel-type zero-free regions for L-functions." pith.science (2026). https://pith.science/paper/Q5KTNOLY

@misc{pith2026260812257,
  author       = {Pith},
  title        = {Pith review of: Power sums and Siegel-type zero-free regions for L-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q5KTNOLY}},
  note         = {Machine review of arXiv:2608.12257}
}
abstract

Let $\pi$ and $\pi'$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ over a number field $F$. Let $\mathfrak{C}_{\pi}$ be the analytic conductor of $\pi$. We develop a new approach to zero-free regions for $L$-functions via lower bounds for power sums, proving for all $\varepsilon>0$ the existence of ineffective constants $c=c_{n,F,\varepsilon}>0$ and $c'=c'_{n,F,\pi',\varepsilon}>0$ such that the standard $L$-function $L(s,\pi)$ satisfies \[ |L(\sigma+it,\pi)|\geq c(\mathfrak{C}_{\pi}(|t|+3))^{-\varepsilon},\qquad \sigma\geq 1-c(\mathfrak{C}_{\pi}(|t|+3))^{-\varepsilon} \] and the Rankin-Selberg $L$-function $L(s,\pi\times\pi')$ satisfies \[ |L(\sigma+it,\pi\times\pi')|\geq c'(\mathfrak{C}_{\pi}(|t|+3))^{-\varepsilon},\qquad \sigma\geq 1-c'(\mathfrak{C}_{\pi}(|t|+3))^{-\varepsilon}. \] Applications include improvements to the prime number theorems for these $L$-functions and new generalizations of the Brauer-Siegel theorem.

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