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On the Effective Non-vanishing of Rankin--Selberg $L$-functions at Special Points

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Pith's one-line read A positive proportion of Rankin–Selberg special L-values, governed by an explicit Euler product, are shown not to vanish.

desk verdict Solid effective refinement of Luo's non-vanishing theorem; the main new results are genuine and the cited large sieve is not load-bearing. read the letter →

arxiv 2506.08546 v3 pith:7GB5MXVE submitted 2025-06-10 math.NT

Zhi Qi This is my paper
classification math.NT MSC 11M4111F72
keywords Rankin–SelbergL-functionsnon-vanishingspectralmomentsKuznetsovtraceformulamollificationHecke–MaasscuspformsEulerproductshortintervals
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, for any fixed holomorphic Hecke newform Q of square-free level, a positive proportion of the Rankin–Selberg L-values attached to the Hecke–Maass spectrum do not vanish at their special points: at least γ(Q)/11 of the values L(1/2+it_j, Q⊗u_j) with t_j ≤ T are nonzero as T→∞, where γ(Q) is an explicit Euler product depending only on Q. On short spectral intervals |t_j−T| ≤ T^μ with 3/4<μ<1, the non-vanishing proportion is at least γ(Q)(4μ−3)/(4μ+7−ε). These are effective, form-dependent positive-proportion results, refining earlier asymptotic first-moment and qualitative positive-proportion results. The proof works through asymptotic formulae for the twisted first and second spectral moments of these L-values, using the Kuznetsov trace formula, Voronoï summation, stationary-phase analysis, and a Selberg-type mollifier whose coefficients are chosen optimally.

What carries the argument

The central objects are the twisted spectral moments C_1(m) and C_2(m_1,m_2) with Gaussian weight exp(−(t_j−T)^2/$Π^{2}$), evaluated through the Kuznetsov trace formula, the Voronoï summation formula, and stationary-phase analysis of the resulting Bessel integrals. The first moment has its main term only at m=1; the second moment carries the main term through the Rankin–Selberg L-function L(s,Q⊗Q) expanded near its pole at s=1, producing the constants γ_1, γ_0 and the factor B(m). The mollifier M_j = Σ_{m≤M} x_m a(m) λ_j(m) $m^{{−1/2−it_j}}$ is chosen by optimizing a Rayleigh quotient with coefficients x_m, and the key input controlling the second mollified moment is the mean Lindelöf bound obtained from Luo's spectral large sieve inequality.

What would settle it

Compute numerically, for a fixed Q of small level such as an elliptic curve newform, the proportion #{j: t_j ≤ T, L(1/2+it_j, Q⊗u_j) ≠ 0} / #{j: t_j ≤ T} for increasingly large T; if it ever drops below γ(Q)(1/11−ε), Theorem 1 is false. More directly, test the spectral large sieve inequality (8.3) with a_n=1 for N ~ T on spectral intervals of length 1; a counterexample at that level would destroy the proof of Theorem 8.

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Extended reading notes

Core claim

This paper establishes that, for a fixed holomorphic Hecke newform Q of square-free level q, the special Rankin–Selberg L-values L(1/2+it_j, Q⊗u_j) — where the u_j run over an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue 1/4+$t_j^{2}$ — vanish less often than previously known: at least γ(Q)(1/11−ε) of the values with t_j ≤ T are nonzero as T→∞, with γ(Q) equal to the explicit Euler product in (1.5) built from the Hecke eigenvalues a(p). For short windows |t_j−T| ≤ T^μ with 3/4<μ<1, the non-vanishing proportion is at least γ(Q)(4μ−3)/(4μ+7−ε). The proof proceeds by asymptotically evaluating the twisted first and second spectral moments of these L-values, then applying a Selberg-type mollifier to convert the moment information into a counting lower bound.

Load-bearing premise

The argument depends on the spectral large sieve inequality (8.3): that sums over nearby Maass forms of arbitrary coefficient vectors are as small as (T+N)(TN)^ε, and if that inequality fails, the mean Lindelöf bound and the entire mollified second-moment control collapse.

Editorial extensions

If this is right

  • For each fixed Q, the lower bound γ(Q)/11 is explicit and computable from the Hecke eigenvalues a(p), so the result yields a concrete numerical proportion for any given newform.
  • On short spectral intervals of length about T^μ with μ>3/4, the non-vanishing proportion stays bounded below by γ(Q)(4μ−3)/(4μ+7), so the phenomenon is not confined to the full spectral range.
  • The asymptotic for the untwisted second spectral moment gives a power-saving remainder, refining Luo's smoothed moment asymptotics.
  • In the Phillips–Sarnak deformation picture, a positive proportion of non-vanishing special values is exactly what forces the Weyl law to fail for generic co-finite groups under the standard eigenvalue multiplicity assumptions, as recalled in the introduction.
  • The same moment asymptotics determine the form Q: if |L(s_j,Q⊗u_j)| = c |L(s_j,Q'⊗u_j)| for all j with fixed c>0 then Q=Q'.

Reading between the lines

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

  • A natural testable extension is to evaluate γ(Q) numerically for a few concrete newforms (for instance, the Δ form or a CM form) and compare the observed non-vanishing proportion with γ(Q)/11; the paper itself notes that γ(Q) requires numerical evaluation.
  • The short-interval constant (4μ−3)/(4μ+7) is small even at μ close to 1; pushing the method below μ=3/4 would likely require a genuinely new bound for the second spectral moment rather than a sharper analysis of the present argument.
  • Theorem 10 suggests that the twisted second moment at the points t_j is a complete invariant for the newform Q; a stronger version might establish the same conclusion using only a finite set of exceptional j.
  • The load-bearing role of the spectral large sieve inequality indicates that any improvement of that inequality with N ~ T would translate directly into an improvement of the 1/11 proportion.
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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

0 major / 5 minor

Summary. The paper proves an effective positive-proportion non-vanishing result for the Rankin--Selberg L-values L(1/2+it_j, Q⊗u_j), where Q is a holomorphic Hecke newform of square-free level and u_j are Hecke--Maass cusp forms of full level. Theorem 1 states that, as T→∞, the unweighted proportion of non-vanishing values among 0<t_j≤T is at least γ(Q)(1/11−ε), with γ(Q) given by an explicit Euler product in (1.5). Theorem 2 gives the analogous lower bound γ(Q)(4μ−3)/(4μ+7)−ε on short intervals |t_j−T|≤T^μ for 3/4<μ<1, and Theorem 3 gives a weaker power bound for 1/3<μ≤3/4. The proof develops asymptotic formulae for twisted first and second spectral moments (Theorems 4 and 5), obtains a mollified first and second moment (Lemmas 10.1 and 10.3), and converts the harmonic-weighted lower bound into an unweighted statement by the standard Kowalski--Michel/BHS weight-removal argument. An additional section derives unsmoothed moment asymptotics (Theorem 11) and a determination theorem (Theorem 10).

Significance. If the proof is correct, this is a substantial quantitative advance over Luo's earlier qualitative positive-proportion result: the proportion is given by an explicit, Q-dependent Euler product, and the method also yields short-interval non-vanishing and improved unsmoothed moment asymptotics. The main chain of the argument is coherent: the central Theorems 1 and 2 follow from the mollified-moment asymptotics, which in turn rest on the twisted-moment theorems proved in Sections 6--7 rather than on the spectral large sieve used only for Theorem 3. The paper is careful with constants and derives γ(Q) from residues of Rankin--Selberg L-functions, with no fitted free parameters. The twisted-moment analysis is long and technical, but the structure of lemmas and error terms is clear, and the external inputs (Kuznetsov formula, Voronoï summation, Luo's identity, and the standard weight-removal method) are standard in the field.

minor comments (5)
  1. [§10.4, final paragraph] The passage from the harmonic-weighted lower bound (10.37)--(10.38) to the unweighted statements of Theorems 1 and 2 is delegated to the method of Kowalski--Michel [KM1] and its Maass-form adaptation in [BHS]. Since the precise constant γ(Q)/11 is a headline feature, the authors should either state the exact weight-removal lemma they are invoking or give a precise reference to the theorem in [BHS] that preserves the constant, so that the reader does not have to infer it from the literature.
  2. [§7.3.5] The proof of the off-diagonal second moment says that only the generic case (c,k1k2)=1 will be treated and that the remaining To-type exponential sums are 'not hard' in view of Lemma 7.7. For a paper of this length and with this level of detail elsewhere, a few sentences or a short appendix describing the non-generic cases would greatly help the reader verify that the bound in Lemma 7.12 indeed covers all c, k1, k2.
  3. [§8.1] The spectral large sieve (8.3) is quoted from Luo and Jutila without proof. This is acceptable as an external input, but the paper should state more explicitly in Section 8 that this input is used only for Theorem 3 and for the mean Lindelöf bound (2.8), and not for the central positive-proportion Theorems 1 and 2.
  4. [§10.3, (10.3)] The stated coefficient bound x_m=O(τ(m)) does not strictly follow from (10.26), since 1/ξ(m) can be as large as 3^{ω(m)} for square-free m with many small prime factors. All later estimates in the paper only require x_m=O(T^ε) for m≤M, so the statement of (10.3) should be adjusted accordingly.
  5. [Theorem 4 and §6.4] The notation '3√' (or '3 a') for the cube root in the error term of (2.3) is easy to misread as a product involving a factor 3; the authors should explicitly define the cube-root notation at its first occurrence.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the non-vanishing constants and proportions are derived from independent moment asymptotics, not fitted to the target.

full rationale

The central claim (Theorem 1, equation (1.4)) follows from the mollified-moment lower bound (10.33), which combines Lemma 10.1 (first moment, from Theorem 4) and Lemma 10.3 (second moment, from Theorem 5). Theorem 4 is proved in Section 6 via the Kuznetsov formula, Voronoi summation, and the Wilton bound; Theorem 5 is proved in Section 7 via the Kuznetsov formula, Luo's identity, and the stationary phase analysis of Bessel integrals. The constants gamma_1 and gamma_5 are residues or Euler products attached to L(s, Q tensor Q) and D(s, Xi) (Definitions 3.1 and 10.1), computed from (3.31) and (10.22)-(10.24), and the quotient gamma = gamma_5 / gamma_1 is algebraically identical to the explicit Euler product (1.5). No parameter is fitted to the non-vanishing density; the mollifier coefficients are chosen by Cauchy optimization (10.21), and the resulting proportion depends only on the Euler product. The spectral large sieve (8.3) of Luo and Jutila is used only in Theorem 8, which supports the short-interval Theorem 3 and the untwisted mean Lindelof bound, and it does not enter the proofs of Theorems 4, 5, or 10.1. The cited unsmoothing and harmonic-weight removal steps (Lemma 9.1 from [LQ] adapted from [IJ]; Kowalski-Michel and BHS) are standard external reductions and do not assume the target non-vanishing statement. The few self-citations, such as [LQ], are auxiliary lemmas from prior work and are not constructs that presuppose Theorem 1 or Theorem 2.

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

The central claim is proved from standard tools of analytic number theory: the Kuznetsov trace formula, Voronoï summation, the Deligne and Wilton bounds, and the spectral large sieve of Luo and Jutila. No numerical parameter is fitted to data; γ(Q) is an explicit Euler product depending on the fixed form Q. No new physical or mathematical entities are introduced.

assumptions (6)
  • standard math Kuznetsov trace formula (Lemma 3.1)
    Used to transform the spectral sums into Kloosterman sums and diagonal terms; see §6.1 and §7.1.
  • standard math Voronoï summation for GL(2) (Lemma 3.3)
    Applied to the n-sums in the off-diagonal analysis in §7.3.7.
  • standard math Spectral large sieve inequality of Luo and Jutila (8.3)
    Critical for the mean Lindelöf bound in Theorem 8, which controls the mollified second moment.
  • standard math Deligne bound |a_p(n)| ≤ τ(n) (3.19)
    Used throughout for bounding Hecke eigenvalues of Q.
  • standard math Wilton bound for sums of a(n)e(γn) (Lemma 3.4)
    Used to estimate the Wilton exponential sums in §6.3.3.
  • domain assumption Method of Kowalski and Michel for removing the harmonic weight, as adapted in [BHS]
    Used in §10.4 to convert weighted non-vanishing proportions to unweighted ones; not proved in this paper.

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Pith. "Pith review of On the Effective Non-vanishing of Rankin--Selberg $L$-functions at Special Points." pith.science (2026). https://pith.science/paper/7GB5MXVE

@misc{pith2026250608546,
  author       = {Pith},
  title        = {Pith review of: On the Effective Non-vanishing of Rankin--Selberg $L$-functions at Special Points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7GB5MXVE}},
  note         = {Machine review of arXiv:2506.08546}
}
abstract

Let $Q (z)$ be a holomorphic Hecke cusp newform of square-free level and $u_j (z)$ traverse an orthonormal basis of Hecke--Maass cusp forms of full level. Let $1/4 + t_j^2$ be the Laplace eigenvalue $u_j (z)$. In this paper, we prove that there is a constant $ \gamma (Q) $ expressed as a certain Euler product associated to $Q$ such that at least $ \gamma (Q) / 11 $ of the Rankin--Selberg special $L$-values $L (1/2+it_j, Q \otimes u_j)$ for $ t_j \leqslant T$ do not vanish as $T \rightarrow \infty$. Further, we show that the non-vanishing proportion is at least $\gamma (Q) \cdot (4\mu-3) / (4\mu+7) $ on the short interval $ |t_j - T| \leqslant T^{\mu} $ for any $3/4 < \mu < 1$.

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  1. On the Effective Non-vanishing of Hecke--Maass $L$-functions at Special Points

    math.NT 2025-07 conditional novelty 7.0 of 10

    At least 33% of the special values L(1/2+it_f, f) of Hecke-Maass L-functions are nonzero, a new effective non-vanishing record.

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