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Polynomials in derivatives of automorphic L-functions obey an explicit zero-counting law.

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2026-08-03 13:55 UTC pith:WQM2PLWM

load-bearing objection The framework is genuinely new and worth knowing, but Theorem 1.1 as stated is false: deg_cond is defined using q_π while the proof uses log q_π, so a Dirichlet L-function of conductor q>1 already contradicts the main term. the 3 major comments →

arxiv 2512.22451 v2 pith:WQM2PLWM submitted 2025-12-27 math.NT

Zeros of Polynomials in Derivatives of Automorphic L-functions

Anji Dong , Nawapan Wattanawanichkul , Alexandru Zaharescu This is my paper
classification math.NT MSC 11F6611M26
keywords automorphic L-functionsderivatives of L-functionszeros of L-functionsRiemann–von Mangoldt formulacritical lineBohr–Landau phenomenontrivial zerosfunctional equation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves that every function F(s,π) built from finitely many cuspidal automorphic L-functions and their derivatives satisfies the same kind of zero-counting asymptotic that is classical for the Riemann zeta function: the number of nontrivial zeros up to height T is α₁ T log T + α₂ T + O_F(log T). The constants are explicit: α₁ is the maximal rank-weighted degree of the polynomial, and α₂ depends on the conductor-weighted degree, the rank, and the first nonzero coefficient of the Dirichlet series. The proof works by establishing an approximate functional equation that isolates the dominant monomials and then counting zeros via a contour integral. A second theorem shows that under a modest mean-square bound for the component L-functions on the critical line, almost all nontrivial zeros of any such F lie arbitrarily close to Re(s)=1/2.

Core claim

The central discovery is that the entire algebra B — polynomials with complex coefficients in arbitrary derivatives of cuspidal automorphic L-functions — has a universal zero-counting behavior controlled by three simple invariants: the rank-weighted degree, the conductor-weighted degree (formed from the arithmetic conductors of the components), and the first index of a nonzero Dirichlet coefficient. Theorem 1.1 gives the explicit asymptotic N_F(0,T)=α₁T log T+α₂T+O_F(log T), with α₁=(1/2π)deg_rk(F) and α₂=(1/2π)(deg_cond(F)−deg_rk(F)log(2πe)−log n_F), for all F whose dominant monomials (index set J) have coefficients summing to a nonzero number. Theorem 1.2 asserts that, whenever each compon

What carries the argument

The load-bearing object is the asymptotic functional equation (Lemma 3.3): for s in a half-plane away from the lattice of Gamma-pole points, F(1−s, eπ) is expressed as a main term — a sum over the dominant index set J of products of the original L-functions, Gamma factors, cosine factors, and logarithmic factors B(s,l,π_u) — times (1+O(1/log s+e^{−κ₁s})). This equation transfers the zero problem to the zeros of a finite product of cosine and Gamma factors, allowing Rouché's theorem to place the trivial zeros in small disks around the points s=2n−1+μ_{π_u}(r) and reducing the nontrivial count to a contour integral of the logarithmic derivative, evaluated through a Hadamard-product representat

Load-bearing premise

The counting argument assumes the dominant main term in the functional equation has no zeros in the small disks where the trivial zeros are located; the paper justifies this by saying 'every factor is nonzero' even though that term is a sum, and a cancellation there would break the argument.

What would settle it

For a primitive Dirichlet character χ of conductor q>1, take F(s)=L(s,χ)+L′(s,χ) and numerically count its nontrivial zeros up to large T; the claimed Theorem 1.1 with the stated deg_cond predicts α₂=(q−log(2πe))/2π, while the proof's (5.26) would yield α₂=(log q−log(2πe))/2π, so the two differ by (q−log q)T/2π and the numerical count decides which constant (if either) governs the main term.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The same asymptotic applies to the a-points of any F in B: the zeros of F(s,π)−a are counted by Theorem 1.1 with the same α₁ and a possibly shifted n_{F−a}, recovering classical a-point results for ζ and its derivatives.
  • Specializing to ζ(s), ζ^(k)(s), and polynomials in zeta derivatives recovers the classical zero-counting theorems of the subject, so the paper unifies them in one result.
  • Under the second-moment assumption, the classical result that almost all zeros of ζ(s) lie near the critical line extends to every F in B, giving an O_F(T log log T/δ) bound for the exceptional count.
  • Finite truncations of the Taylor expansion of a shifted L-function, being polynomials in derivatives, have almost all zeros near Re(s)=1/2, even though the full shifted L-function has zeros on a different vertical line.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The proof of Lemma 4.2 justifies A₁≠0 by saying 'every factor of A₁ is nonzero,' but A₁ is a sum; a cancellation among dominant terms is logically possible and would leave the trivial-zero counting without a foundation unless a separate non-vanishing argument is supplied.
  • The constants in Theorem 1.1 as stated use deg_cond defined with q_π (no logarithm), while the proof at (5.26) writes exp(−iT·deg_cond) after computing Σ log q_π; for a conductor q>1 the two readings differ by (q−log q)T/2π in the main term, so the correct interpretation of deg_cond needs to be fixed.
  • If the asymptotic is correct, it gives a practical probe of arithmetic data: the coefficient α₂ encodes the conductor-weighted degree, so numerical zero counts of a known polynomial could, in principle, be used to estimate the conductor of an unknown component.
  • The second-moment assumption (1.9) is stated per component; a natural weakening would be to replace it by a Lindelöf-type bound or by moment estimates for the derivatives on the critical line, which would make Theorem 1.2 unconditional in more families.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper defines a class B of polynomials in derivatives of automorphic L-functions and proposes two main results. Theorem 1.1 claims an asymptotic Riemann–von Mangoldt type formula for the number of nontrivial zeros of F(s,π) with 0<Im(s)<T, with explicit constants α1 and α2 expressed in terms of rank, derivative order, and arithmetic conductor degrees defined in (1.2)–(1.4). Theorem 1.2 claims that, under a second-moment bound (1.9), almost all nontrivial zeros of such F(s,π) lie arbitrarily close to the critical line Re(s)=1/2. The proof strategy is to establish an asymptotic functional equation (Lemma 3.3), locate trivial zeros via Rouché's theorem (Proposition 4.1 and Lemma 4.2), and then perform a contour integration argument in Section 5. Applications to derivatives of zeta, polynomials in zeta derivatives, a-points, and truncated Taylor series are discussed.

Significance. If the theorems were correct, they would unify and extend a large body of zero-counting results for automorphic L-functions and their derivatives, with explicit constants generalizing the classical Riemann–von Mangoldt formula. The paper also offers a conditional clustering statement near the critical line. These are valuable goals. However, the central Theorem 1.1 is internally inconsistent: the conductor degree defined in (1.4) is Σ q_π, while the proof repeatedly uses Σ log q_π. This makes the stated theorem false for a single Dirichlet L-function of conductor q>1. Additionally, the zero-free region and trivial-zero counting arguments in Section 4 contain a logical gap concerning sums of nonzero terms. These are load-bearing flaws, so the paper's main contributions are not established in their current form.

major comments (3)
  1. [§1.1, Eq. (1.4); §5, Eq. (5.26)] The definition of degcond in (1.4) is Σ_u q_{π_u} Σ_l d_{u,l,j}, but equation (5.26) asserts exp(-iT Σ_u log q_{π_u} Σ_l d_{u,l,j}) = exp(-iT degcond(F)), which requires degcond = Σ log q_{π_u}. The same mismatch appears in (3.20) and (5.25). Consequently Theorem 1.1 gives the wrong α2. For a primitive Dirichlet character χ mod q>1, Theorem 1.1 predicts α2=(q-log(2πe))/(2π), whereas the classical formula is α2=(log q - log(2πe))/(2π). This is not a typographical issue; the definition of the conductor degree must be changed throughout, and all conclusions depending on it recomputed.
  2. [Proposition 4.1] The proof asserts that A1(s,eπ) is nonzero because 'every factor of A1 is nonzero'. But A1 is a sum over j∈J of products, and a sum of nonzero terms can vanish through cancellation. Condition (1.7) only fixes the coefficient sum, not the uniform nonvanishing of the main term. Without a lower bound or an additional dominance argument, the claimed zero-free region for σ large is not established, and the subsequent separation of trivial and nontrivial zeros collapses.
  3. [Lemma 4.2] The zero count for A1 inside each disk K_{n,u,r} is unjustified. The assertion that 'the zeros of A1 can only arise from the product of cosine terms' is false for a sum: the zeros of Σ c_j ∏ ... are not controlled by the zeros of individual factors. In particular, cancellation can create zeros away from the points 2n-1+μ_{π_u}(r). Even if each disk contained exactly one simple zero, the union K_n would contain at most degrk(F) zeros only if the disks are disjoint and no other zeros appear; none of this is proved. This invalidates the definition of N_{F} used in the contour argument.
minor comments (4)
  1. [§3, Eq. (3.5)] The summation condition '0≤k4,r,k4,r≤ku' should presumably be '0≤k4,r,k5,r≤ku'. Please correct the typo.
  2. [§4, Proposition 4.1 / Lemma 4.2] The punctured regions are written with a union over n of sets; the interplay with the disks C_n and possible overlaps of disks for different u,r is not discussed. Clarify the counting when disks overlap.
  3. [§6, Lemma 6.1] The constant D is referred to as 'taken from in the proof of Lemma 5.2' but not defined at the start of Lemma 6.1. Define D explicitly before use.
  4. [§3, Lemma 3.3] The error term '1/log s + e^{-κ1 s}' should specify the branch of log and the dependence on s in the bound; as written it is ambiguous when s is near a negative real or has large imaginary part.

Circularity Check

0 steps flagged

No significant circularity: the zero-counting and critical-line theorems are derived from stated hypotheses and external functional equations, not from the results they prove.

full rationale

The paper's derivation chain is self-contained rather than circular. Theorem 1.1 is obtained by (i) an asymptotic functional equation for F(1-s, eπ) derived from the standard automorphic L-function functional equation (Lemma 3.3); (ii) zero-free regions and trivial-zero location from that asymptotic equation (Proposition 4.1, Lemma 4.2); and (iii) a Hadamard-product/log-derivative argument plus argument principle (Lemmas 5.1–5.3). The constants α1 and α2 are computed, not fitted: no parameter is matched to the N_F(0,T) being counted, and condition (1.7) is a nondegeneracy assumption rather than an input from which the asymptotic is forced. Theorem 1.2 explicitly states the strong second-moment hypothesis (1.9) as an assumption and derives the conclusion from it; an independent hypothesis used as input is not circular. There are no load-bearing self-citations: the paper cites Berndt, Levinson–Montgomery, Onozuka, Rudnick–Sarnak, etc., as external checks or standard tools, not as the source of the central derivation. The recoveries of classical results in Section 1.3 are consequences of the theorem, not ingredients of its proof. I note for completeness two serious non-circular issues: (1) deg_cond is defined in (1.4) using q_{π_u}, whereas the proof's equation (5.26) replaces Σ log q_{π_u} by deg_cond(F(s,π)), an identity that is false as written; this makes the stated α2 inconsistent with the classical Dirichlet L-function case, but it is a correctness/falsity problem, not a circular one. (2) Proposition 4.1 asserts A1(s,eπ)≠0 because every factor is nonzero, although A1 is a sum; again, this is an unsupported proof step, not a self-referential use of the theorem. Because the paper never assumes the asymptotic it proves, no circular step is present, and the appropriate circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No free parameters are fitted; the constants α₁, α₂ are derived. The main exogenous inputs are the standard theory of automorphic L-functions and the explicit analytic hypotheses (second moment, RH/H_π). The latter are honest assumptions, not hidden fits.

axioms (6)
  • standard math Standard analytic properties of cuspidal automorphic L-functions on GL_m(A_Q): Euler product, analytic continuation, functional equation with gamma factors and conductor q_π (Section 2).
    These are established theorems from the theory of automorphic forms; the paper takes them as input.
  • domain assumption Ramanujan–Selberg-type bounds θ_m ≤ 1/2 − 1/(m^2+1) from Luo–Rudnick–Sarnak, Müller–Speh, Kim–Sarnak, Blomer–Brumley (eq. 2.4).
    Used in Lemma 3.1 to bound Dirichlet coefficients; these are proven but deep results.
  • domain assumption Strong second-moment bound (1.9): ∫|L(1/2+it,π_u)|² dt ≪ T (log T)^η for each component.
    Explicit hypothesis of Theorem 1.2; unknown in full generality.
  • domain assumption Hypothesis A (RH for certain L-functions plus Hypothesis H_π) in Corollary 1.4.
    Unproved Riemann Hypothesis for the relevant L-functions.
  • ad hoc to paper Nonvanishing of the leading coefficient sum (1.7): Σ_{j∈J} c_j ≠ 0.
    Technical condition needed for the dominant term not to cancel.
  • domain assumption Hypothesis H_π: Σ_p |λ_π(p^k)|²/p^k < ∞ (Rudnick–Sarnak), stated as an assumption throughout although the text notes Jiang established it.
    Used to ensure analytic properties; cited to a recent preprint.

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0 comments
read the original abstract

Let $\mathfrak{F}_m$ be the set of all cuspidal automorphic representations of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$, and let $F(s,\boldsymbol{\pi})$ be a polynomial in the derivatives of $L$-functions associated with representations $\pi \in \cup_{m=1}^{\infty} \mathfrak{F}_m$. We establish an asymptotic formula for the number of nontrivial zeros of $F(s,\boldsymbol{\pi})$ with $0 < \operatorname{Im}(s) < T$. We explicitly determine the main term of this formula in terms of the dimensions, the arithmetic conductors, and the orders of differentiation of the component $L$-functions. Furthermore, we show that, under certain conditions, almost all nontrivial zeros of $F(s,\boldsymbol{\pi})$ lie near the critical line $\operatorname{Re}(s)=1/2$.

Figures

Figures reproduced from arXiv: 2512.22451 by Alexandru Zaharescu, Anji Dong, Nawapan Wattanawanichkul.

Figure 1
Figure 1. Figure 1: The rectangular contour and the encompassing quarter-circular arc Putting together, we have that F ′ (E + it,π) F(E + it,π) ≪ 1. (5.14) Now taking s = E + it in (5.1) from Lemma 5.1, we obtain F ′ (E + it,π) F(E + it,π) = pF 1 − E − it + zF E + it + BF + X ρF ̸=0  1 E + it − ρF + 1 ρF  . (5.15) Subtracting (5.15) from (5.1) and using (5.14), we obtain that for σ1 < σ < σ2 and large t, F ′ (s,π) F(s,π) = … view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

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