Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

A zero-density estimate for $L$-functions associated with $\rm GL(3)$ Hecke--Maass cusp forms

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read On a spectrally localized family of GL(3) Hecke–Maass cusp forms, almost no L-functions have zeros to the right of $1/2+\sigma$ up to height $H$, for $H$ as small as $3/\log T$.

desk verdict A technically serious paper whose stated main theorem is not proven as written: Section 7 conflates the box height H with the spectral normalizer H_spec, and the small-H end of the uniform range is not controlled. read the letter →

arxiv 2412.02416 v1 pith:5N2MFEPK submitted 2024-12-03 math.NT

Qingfeng Sun , Hui Wang This is my paper · ORCID
classification math.NT MSC 11F1211F6611F6711F72
keywords GL(3)L-functionsHecke–Maasscuspformszero-densityestimatespectralaspectKuznetsovtraceformulatwistedsecondmomentmollificationargumentprinciple
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 establishes a weighted zero-density estimate for the $L$-functions attached to $\mathrm{GL}(3)$ Hecke–Maass cusp forms on $\mathrm{SL}(3,\mathbb{Z})$. If the theorem is correct, then on average over a spectrally localized family of such forms, essentially no $L$-function has a zero with real part to the right of $1/2+\sigma$ and imaginary part below $T^{\theta_1}$, for any fixed $\sigma>0$ that is not too small. The proof goes through a new asymptotic formula for the twisted second moment of these $L$-functions, proved by means of the $\mathrm{GL}(3)$ Kuznetsov trace formula, together with a mollified second-moment bound and the argument-principle lemma. The result extends to the spectral aspect of $\mathrm{GL}(3)$ a line of zero-density estimates previously known for $\zeta(s)$ and for $\mathrm{GL}(2)$ forms, and the authors note it can replace the generalized Riemann hypothesis in applications such as the study of $S(t)$ for $\mathrm{GL}(3)$.

What carries the argument

The engine of the proof is the Kuznetsov trace formula on $\mathrm{SL}(3,\mathbb{Z})$ (Lemma 3.2), applied with the test function $h_{T,M}(\mu)$ that localizes the spectral parameters of the Hecke–Maass forms to a ball of radius $M=T^{\theta}$ about the generic point $\pm\mu_0$ with $|\mu_{0,k}|\asymp T$. This converts spectral averages of $A_j(\ell_1,\ell_2)|L(1/2+\sigma+i\tau,\varphi_j)|^2$ into diagonal terms plus Kloosterman and Eisenstein contributions, producing the twisted second-moment asymptotic (Theorem 1.2) with main terms $\zeta(1+2\sigma)$ and $\zeta(1-2\sigma)$ times the spectral integral, and error terms of sizes $T^{9/4-6\sigma}M^2$, $T^{2-3\sigma+\varepsilon}M^2$, and $T^{83/42-41\sigma/21+\varepsilon}M^2(\ell_1\ell_2)^{\vartheta}$. The mollified second moment (Proposition 1.4) is then bounded by $1+O(T^{-2\sigma\delta})$ using a mollifier of length $L=T^\delta$. Finally, the argument-principle lemma (Lemma 4.1) with the box $1/2+1/\log T \pm 2iH$ to $3/2\pm 2iH$ converts this bound into the zero-density estimate, because only the real part of $\log |LM(s,\varphi_j)|$ inside the critical strip needs to be controlled.

What would settle it

Keep every normalization explicit when Proposition 1.4 is substituted into the argument-principle inequality in Section 7; if the unsuppressed spectral factor $H_{\mathrm{spec}}\asymp T^3M^2$ remains after multiplying by the box height $H$, the stated bound $N(\sigma,H)\ll H T^{-\delta\sigma}\log T$ cannot follow, and a corrected normalization or a different right-hand side would be needed.

Watch

Extended reading notes

Core claim

Theorem 1.1 is the paper's central claim. For $2/\log T < \sigma < 1/2$ and for sufficiently small $\delta,\theta_1>0$, the weighted zero count defined in (1.3) obeys $$N(\$\sigma$,H) \ll H $T^{{-\delta\sigma}}$\log T \qquad\text{uniformly for }3/\log T<H<$T^{{\theta_1}}$.$$ Here $N(\sigma,H)$ averages, over the $h_{T,M}$-localized harmonic family, the number of zeros with real part exceeding $1/2+\sigma$ and imaginary part in $(-H,H)$, divided by the box height $H$. The paper's own gloss (Remark 1) is that there are very few $L$-functions in this family with zeros to the right of $1/2+C/\log T$ and below height $T^{\theta_1}$.

Load-bearing premise

The proof assumes that the factor $1/H$ in the definition of the weighted zero count and the spectral normalization factor in the mollified second-moment bound are compatible, without showing how the large spectral factor $H_{\mathrm{spec}}\asymp T^3M^2$ is removed in the final step of Section 7.

Editorial extensions

If this is right

  • For any fixed $\sigma>0$, the weighted average number of zeros with real part $\ge 1/2+\sigma$ and imaginary part below $T^{\theta_1}$ tends to zero as $T\to\infty$, so the family is asymptotically zero-free in that box in the harmonic-average sense.
  • The bound is uniform in $H$ down to $3/\log T$, the scale on which the total number of zeros is only $O(\log T)$, so the estimate remains useful near the critical line.
  • Letting $\sigma\to0$ in the twisted second moment (Corollary 1.3) yields an asymptotic for the harmonic average of $A_j(\ell_1,\ell_2)|L(1/2+i\tau,\varphi_j)|^2$, a tool for non-vanishing and moment problems for GL(3) L-functions.
  • As the authors note, the weighted zero-density estimate can replace the generalized Riemann hypothesis in arithmetic applications such as the GL(3) analogue of the $S(t)$ problem.

Reading between the lines

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

  • Beyond the paper: the same template—Kuznetsov formula, twisted second moment, mollification, argument principle—should carry over to other automorphic families, such as GL(3) forms of varying level or weight, once the analogous moment asymptotics are available.
  • Beyond the paper: because the bound is harmonic-weighted, it does not by itself control the proportion of individual forms with an exceptional zero; a non-weighted statement would require an additional large-sieve or spectral-density argument.
  • Beyond the paper: the uniformity down to $H \asymp 3/\log T$ makes the estimate sensitive to the true density of low-lying zeros, so comparing the implied count with the expected number from a GL(3) Weyl law would test whether the exponent $\delta\sigma$ has the right shape.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a Kuznetsov-based approach to the spectral second moment of GL(3) Hecke–Maass L-functions. It states an asymptotic formula (Theorem 1.2) for the twisted second moment with the localizing weight h_{T,M}, derives from it a mollified second moment bound (Proposition 1.4), and then uses a Selberg/Conrey–Soundararajan argument-principle lemma to claim a weighted zero-density estimate (Theorem 1.1) of the form N(σ,H) ≪ H T^{-δσ} log T, uniformly in 3/log T < H < T^{θ1}. The proof follows the standard route: approximate functional equations, the GL(3) Kuznetsov trace formula, character sum estimates, Eisenstein contribution bounds via subconvexity, and contour shifts for the mollified moments.

Significance. If Theorem 1.1 were valid, it would be a first spectral-aspect zero-density estimate of Selberg type for GL(3) automorphic L-functions, and the twisted/mollified second moment calculations in Theorems 1.2 and Proposition 1.4 would be valuable technical ingredients. The paper is largely parameter-free: the main terms come from residues, the test function is explicit, and I saw no circular use of the target result or fitted constants. The arithmetic sections appear carefully executed and cite the relevant subconvexity and spectral tools. However, the central deduction in Section 7 contains a normalization mismatch between the box height in (1.3) and the spectral normalizer in (1.8), and a separate horizontal-edge issue affects the small-H range. As a result, the advertised theorem does not follow from the established estimates. The component results may still be of interest, but the paper's central claim is unsupported.

major comments (2)
  1. [Section 7; Eqs. (1.3), (1.7)–(1.8)] The proof of Theorem 1.1 conflates the box height H in (1.3) and Lemma 4.1 with the spectral normalizer H defined in (1.8). Proposition 1.4 establishes a bound for (1/H_spec)Σ_j (h_{T,M}/N_j)|LM(1/2+σ+iτ,φ_j)|^2, where H_spec = (1/192π^5)∫ h_{T,M} dμ_spec ≍ T^3 M^2. When this bound is substituted into the displayed estimate in Section 7, the factor 1/H_spec does not cancel with the box height H_box. Keeping the two symbols distinct gives (1/H_box)Σ_j(h/N_j)|LM(·)|^2 ≤ (H_spec/H_box)(1+O(T^{-2σδ})). The t-integral of length H_box then leaves a term of order H_spec log T = O(T^{3+2θ} log T), and no displayed cancellation from Corollary 7.2 removes this factor. Since Theorem 1.1 requires an upper bound O(H T^{-δσ} log T) with H ≤ T^{θ1}, the claimed theorem does not follow from the given estimates. If H in (1.3) is instead intended to be the spectral normalizer, then the box height in N(σ,H;φ_j) is left undefined and the uniformity range in H loses its meaning.
  2. [Section 7; Lemma 4.1 and Remark 7] The horizontal-edge contribution is not controlled in the range H ≈ 3/log T. In Lemma 4.1, the second integral over α has integrand sinh(π(α-W0)/(2H)) log|ω(α± iH)|. In the application, W0 = 1/log T and α runs to 1, so at α = 1 the sinh factor is of size exp(π(1-1/log T)/(2H)) ≍ T^{π/6}. The proof replaces log|x| by (|x|^2-1)/2, so this term requires a bound on the spectral average of |LM(1/2+α+iH)|^2 that would offset this large weight. Proposition 1.4 gives no such decay. Remark 7 asserts that only the real part of the logarithm appears in the part of the integral inside the critical strip, but that observation concerns the vertical side and does not control the horizontal sides. Thus the claimed uniformity in H down to 3/log T is not supported by the argument in the text.
minor comments (4)
  1. [Theorem 1.1, Eq. (1.3)] The statement says “Let 2/log T < σ < 1/2” but the uniformity range is written as 3/log T < H < T^{θ1}; the relation between these thresholds and the choice W0 = 1/log T in Section 7 should be clarified.
  2. [Section 7, first paragraph] The sentence “box bounded by 1/2+1/log T ± 2iH and 3/2 ± 2iH” is inconsistent with the later choice W0 = 1/log T and W1 = 1; a change of variables or a precise definition of the box coordinates should be stated explicitly.
  3. [Section 6, Eq. (6.23)] The phrase “provided for δ ≤ 1/3” is garbled; it should presumably read “provided δ ≤ 1/3”.
  4. [Remark 2] The abstract advertises applications of the zero-density estimate, but Remark 2 defers the proof of the application to the second author's thesis. This makes the claimed relevance non-verifiable in the present paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the zero-density estimate is derived from an independently obtained twisted second moment formula, with no fitted parameters, self-referential definitions, or load-bearing self-citations.

full rationale

The paper's central claim, Theorem 1.1, is derived from Proposition 1.4 and Corollary 7.2. Proposition 1.4 is itself proved from Theorem 1.2, whose proof uses the Kuznetsov trace formula, approximate functional equations, Stirling bounds, and external subconvexity estimates (Bourgain, Meurman, Kim–Sarnak). None of these inputs presupposes the zero-density result. There are no fitted constants disguised as predictions: the only parameter delta enters through the mollifier length L = T^delta, and the final bound's T^{-delta sigma} dependence follows from the mollifier analysis rather than from matching a target exponent. The paper does not rely on a self-citation chain: the cited papers [2, 3, 8] are by Blomer, Buttcane, and Zhou, not by the present authors, and the relevant lemmas are external results with independent proofs or standard trace-formula machinery. The skeptically noted mismatch between the box height H in (1.3) and the spectral normalizer H in (1.8) is a possible substantive mathematical gap in the final step, but it is not a circularity: no theorem or definition is being assumed in the form that is being proved, and the alleged issue is one of normalization/unverified cancellation rather than reduction to input. Accordingly, the circularity score is 0.

Assumptions & free parameters 5 free parameters · 8 assumptions · 0 invented entities

The paper introduces no new particles, mediators, forces, dimensions, or other invented entities. The proof rests on standard GL(3) spectral theory, known subconvexity estimates, and classical zeta-function bounds, all cited from the literature. The free parameters theta, delta, theta1, and A are proof parameters chosen sufficiently small or large; they are not fitted to data and do not introduce circularity.

free parameters (5)
  • theta
    Defines the localization radius M = T^theta of the spectral test function; chosen in (0,1) and sufficiently small for the proof, not fitted to data.
  • delta
    Defines the mollifier length L = T^delta; chosen sufficiently small so the error terms in the mollified second moment are controlled, not fitted to data.
  • theta1
    Exponent bounding the height H < T^{theta1} in Theorem 1.1; chosen sufficiently small, not fitted to data.
  • A
    Degree of the polynomial P(mu) in the test function h_{T,M}; chosen large enough to ensure rapid decay, not fitted to data.
  • epsilon
    Standard arbitrarily small positive exponent used throughout analytic number theory estimates; no fitted value.
assumptions (8)
  • domain assumption GL(3) Kuznetsov trace formula (Buttcane)
    Lemma 3.2 is quoted from Buttcane [7] and is the central spectral decomposition tool; it is a deep external theorem in automorphic forms.
  • domain assumption Kim-Sarnak bound for GL(2) Fourier coefficients, theta = 7/64
    Used in (3.5) to bound coefficients of the maximal Eisenstein series; an external result toward the Ramanujan-Petersson conjecture.
  • domain assumption Subconvexity bounds for zeta and GL(2) Maass L-functions
    Bourgain's bound zeta(1/2+it) << |t|^{13/84+epsilon} and Meurman's bound L(1/2+it,g) << |t|^{1/3+epsilon} are used in Section 5.4 to estimate Eisenstein contributions.
  • domain assumption Functional equation and approximate functional equation for GL(3) L-functions
    Lemma 2.1 is quoted from Iwaniec-Kowalski and is used to expand |L(1/2+sigma+i*tau)|^2 into a sum over Fourier coefficients.
  • standard math Zero-free region and logarithmic bounds for the Riemann zeta function
    Used in Section 6 to shift contours and control the main terms S1 and S2 in the mollified second moment.
  • standard math Argument principle lemma (Selberg, Conrey-Soundararajan)
    Lemma 4.1 is the bridge from second moment bounds to zero counts; stated with proof outline and used directly in Section 7.
  • domain assumption GL(2) Weyl law
    Venkov-Hejhal Weyl law (5.18) is used to sum over the maximal Eisenstein spectrum in Section 5.4.
  • domain assumption Generic spectral assumption |mu_{0,i}| ~ |nu_{0,i}| ~ T
    The theorem is proven away from the Weyl chamber walls and away from self-dual forms; this generic assumption localizes the test function and is stated in the introduction and (2.6).

how reviews work

0 comments
Cite this review

Pith. "Pith review of A zero-density estimate for $L$-functions associated with $\rm GL(3)$ Hecke--Maass cusp forms." pith.science (2026). https://pith.science/paper/5N2MFEPK

@misc{pith2026241202416,
  author       = {Pith},
  title        = {Pith review of: A zero-density estimate for $L$-functions associated with $\rm GL(3)$ Hecke--Maass cusp forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5N2MFEPK}},
  note         = {Machine review of arXiv:2412.02416}
}
abstract

In this paper, we establish an asymptotic formula for the twisted second moment of $L$-functions associated with Hecke--Maass cusp forms for $\rm SL(3,\mathbb{Z})$, and further deduce a weighted zero-density estimate for these $L$-functions in the spectral aspect which may have important applications in other problems.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Selberg-type zero-density result for twisted $\rm GL_2$ $L$-functions and its application

    math.NT 2025-05 conditional novelty 6.0 of 10

    Selberg-type zero-density estimates for twisted GL2 L-functions yield a central limit theorem for the argument function S(t,f⊗χ) as the character modulus q grows.

Reference graph

Works this paper leans on

48 extracted references · 44 canonical work pages · cited by 1 Pith paper

  1. [1]

    Balkanova, B

    O. Balkanova, B. Huang and A. S¨ odergren, Non-vanishing of Maass form L-functions at the central poin t. Proc. Amer. Math. Soc. 149 (2021), no. 2, 509–523

    work page 2021

  2. [2]

    Blomer, Applications of the Kuznetsov formula on GLp3q

    V. Blomer, Applications of the Kuznetsov formula on GLp3q. Invent. Math. 194 (2013), no. 3, 673–729

    work page 2013

  3. [3]

    Blomer and J

    V. Blomer and J. Buttcane, On the subconvexity problem for L-functions on GLp3q. Ann. Sci. ´Ec. Norm. Sup´ er. (4)53 (2020), no. 6, 1441–1500

    work page 2020

  4. [4]

    Bourgain, Decoupling, exponential sums and the Riemann zeta function

    J. Bourgain, Decoupling, exponential sums and the Riemann zeta function . J. Amer. Math. Soc. 30 (2017), no. 1, 205–224

    work page 2017

  5. [5]

    D. Bump, S. Friedberg and D. Goldfeld, Poincar´ e series and Kloosterman sums for SLp3, Zq. Acta Arith. 50 (1988), no. 1, 31–89

    work page 1988

  6. [6]

    Buttcane, On sums of SLp3, Zq Kloosterman sums

    J. Buttcane, On sums of SLp3, Zq Kloosterman sums . Ramanujan J. 32 (2013), no. 3, 371–419. 32 QINGFENG SUN AND HUI W ANG

    work page 2013

  7. [7]

    Buttcane, The spectral Kuznetsov formula on SLp3q

    J. Buttcane, The spectral Kuznetsov formula on SLp3q. Trans. Amer. Math. Soc. 368 (2016), no. 9, 6683–6714

    work page 2016

  8. [8]

    Buttcane and F

    J. Buttcane and F. Zhou, Plancherel distribution of Satake parameters of Maass cusp forms on GL3. Int. Math. Res. Not. IMRN 2020, no. 5, 1417–1444

    work page 2020

Show all 48 references
  1. [9]

    Conrey and K

    J.B. Conrey and K. Soundararajan, Real zeros of quadratic Dirichlet L-functions . Invent. Math. 150 (2002), no. 1, 1–44

    2002

  2. [10]

    Gelbart, E.M

    S.S. Gelbart, E.M. Lapid and P. Sarnak, A new method for lower bounds of L-functions . C. R. Math. Acad. Sci. Paris 339 (2004), no. 2, 91–94

    2004

  3. [11]

    Goldfeld, Automorphic forms and L-functions for the group GLpn, Rq

    D. Goldfeld, Automorphic forms and L-functions for the group GLpn, Rq. With an appendix by Kevin A. Broughan. Cambridge Studies in Advanced Mathematics, 99. C ambridge University Press, Cambridge, 2006. xiv+493 pp

    2006

  4. [12]

    Goldfeld and B

    D. Goldfeld and B. Huang, Super-positivity of a family of L-functions in the level asp ect. Res. Math. Sci. 5 (2018), no. 2, Paper No. 16, 26 pp

    2018

  5. [13]

    Guth and J

    L. Guth and J. Maynard, New large value estimates for Dirichlet polynomials . arXiv: 2405.20552

    arXiv

  6. [14]

    Hal´ asz, ¨Uber die Mittelwerte multiplikativer zahlentheoretische r Funktionen

    G. Hal´ asz, ¨Uber die Mittelwerte multiplikativer zahlentheoretische r Funktionen . (German) Acta Math. Acad. Sci. Hungar. 19 (1968), 365–403

    1968

  7. [15]

    Heath–Brown, Zero density estimates for the Riemann zeta-function and Dir ichlet L-functions

    D.R. Heath–Brown, Zero density estimates for the Riemann zeta-function and Dir ichlet L-functions . J. Lon- don Math. Soc. (2) 19 (1979), no. 2, 221–232

    1979

  8. [16]

    Hejhal, The Selberg trace formula for PSLp2, Rq

    D.A. Hejhal, The Selberg trace formula for PSLp2, Rq. Vol. 2. Lecture Notes in Mathematics, 1001. Springer- Verlag, Berlin, 1983. viii+806 pp

    1983

  9. [17]

    Hoffstein and P

    J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero . With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman. Ann. of Math. (2) 140 (1994), no. 1, 161–181

    1994

  10. [18]

    Hoffstein and D

    J. Hoffstein and D. Ramakrishnan, Siegel zeros and cusp forms . Internat. Math. Res. Notices 1995, no. 6, 279–308

    1995

  11. [19]

    Hough, Zero-density estimate for modular form L-functions in weigh t aspect

    B. Hough, Zero-density estimate for modular form L-functions in weigh t aspect. Acta Arith. 154 (2012), no. 2, 187–216

    2012

  12. [20]

    Huang, S

    B. Huang, S. Liu and Z. Xu, Mollification and non-vanishing of automorphic L-function s on GLp3q. Israel J. Math. 227 (2018), no. 2, 597–622

    2018

  13. [21]

    Huxley, On the difference between consecutive primes

    M.N. Huxley, On the difference between consecutive primes . Invent. Math. 15 (1972), 164–170

    1972

  14. [22]

    Ingham, On the estimation of N pσ, T q

    A.E. Ingham, On the estimation of N pσ, T q. Quart. J. Math. Oxford Ser. 11 (1940), 291–292

    1940

  15. [23]

    Ivi´ c,The Riemann zeta-function

    A. Ivi´ c,The Riemann zeta-function. Theory and applications . Reprint of the 1985 original [Wiley, New York; MR0792089]. Dover Publications, Inc., Mineola, NY, 2003. x xii+517 pp

    1985

  16. [24]

    Iwaniec and E

    H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publica- tions, 53. American Mathematical Society, Providence, RI, 2004. xii+615 pp

    2004

  17. [25]

    Jacquet and J.A

    H. Jacquet and J.A. Shalika, A non-vanishing theorem for zeta functions of GLn. Invent. Math. 38 (1976/77), no. 1, 1–16

    1976

  18. [26]

    Jutila, Zero-density estimates for L-functions

    M. Jutila, Zero-density estimates for L-functions . Acta Arith. 32 (1977), no. 1, 55–62

    1977

  19. [27]

    Khan, Non-vanishing of the symmetric square L-function at the cen tral point

    R. Khan, Non-vanishing of the symmetric square L-function at the cen tral point. Proc. Lond. Math. Soc. (3) 100 (2010), no. 3, 736–762

    2010

  20. [28]

    Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2

    H.H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarna k. J. Amer. Math. Soc. 16 (2003), no. 1, 139–183

    2003

  21. [29]

    Kowalski and P

    E. Kowalski and P. Michel, The analytic rank of J0pqq and zeros of automorphic L-functions . Duke Math. J. 100 (1999), no. 3, 503–542

    1999

  22. [30]

    Kowalski and P

    E. Kowalski and P. Michel, A lower bound for the rank of J0pqq. Acta Arith. 94 (2000), no. 4, 303–343

    2000

  23. [31]

    Kowalski, P

    E. Kowalski, P. Michel and J. VanderKam, Mollification of the fourth moment of automorphic L-functio ns and arithmetic applications . Invent. Math. 142 (2000), no. 1, 95–151

    2000

  24. [32]

    Kowalski, P

    E. Kowalski, P. Michel and J. VanderKam, Non-vanishing of high derivatives of automorphic L-functi ons at the center of the critical strip . J. Reine Angew. Math. 526 (2000), 1–34

    2000

  25. [33]

    Kowalski, P

    E. Kowalski, P. Michel and J. VanderKam, Rankin–Selberg L-functions in the level aspect . Duke Math. J. 114 (2002), no. 1, 123–191

    2002

  26. [34]

    Liu, Nonvanishing of central L-values of Maass forms

    S. Liu, Nonvanishing of central L-values of Maass forms . Adv. Math. 332 (2018), 403–437

    2018

  27. [35]

    Liu and S

    S. Liu and S. Liu, A GL3 analog of Selberg’s result on Sptq. Ramanujan J. 56 (2021), no. 1, 163–181

    2021

  28. [36]

    Liu and J

    S. Liu and J. Streipel, The twisted second moment of L-functions associated to Heck e–Maass forms . Int. J. Number Theory 20 (2024), no. 3, 849–866. 33

    2024

  29. [37]

    Luo, Nonvanishing of L-values and the Weyl law

    W. Luo, Nonvanishing of L-values and the Weyl law . Ann. of Math. (2) 154 (2001), no. 2, 477–502

    2001

  30. [38]

    Luo, Zeros of Hecke L-functions associated with cusp forms

    W. Luo, Zeros of Hecke L-functions associated with cusp forms . Acta Arith. 71 (1995), no. 2, 139–158

    1995

  31. [39]

    Luo, Nonvanishing of the central L-values with large weight

    W. Luo, Nonvanishing of the central L-values with large weight . Adv. Math. 285 (2015), 220–234

    2015

  32. [40]

    Meurman, On the order of the Maass L-function on the critical line

    T. Meurman, On the order of the Maass L-function on the critical line . Number theory, Vol. I (Budapest, 1987), 325–354, Colloq. Math. Soc. J´ anos Bolyai, 51, North –Holland, Amsterdam, 1990

    1987

  33. [41]

    Montgomery, Zeros of L-functions

    H.L. Montgomery, Zeros of L-functions . Invent. Math. 8 (1969), 346–354

    1969

  34. [42]

    Olver, Asymptotics and special functions

    F. Olver, Asymptotics and special functions . Reprint of the 1974 original [Academic Press, New York; MR0435697]. AKP Classics. A K Peters, Ltd., Wellesley, MA, 1 997. xviii+572 pp

    1974

  35. [43]

    Selberg, Contributions to the theory of the Riemann zeta-function

    A. Selberg, Contributions to the theory of the Riemann zeta-function . Arch. Math. Naturvid. 48 (1946), no. 5, 89–155

    1946

  36. [44]

    Selberg, Contributions to the theory of Dirichlet’s L-functions

    A. Selberg, Contributions to the theory of Dirichlet’s L-functions . Skr. Norske Vid.-Akad. Oslo I 1946 (1946), no. 3, 62 pp

    1946

  37. [45]

    Soundararajan, Nonvanishing of quadratic Dirichlet L-functions at s “ 1 2

    K. Soundararajan, Nonvanishing of quadratic Dirichlet L-functions at s “ 1 2 . Ann. of Math. (2) 152 (2000), no. 2, 447–488

    2000

  38. [46]

    Titchmarsh, The theory of the Riemann zeta-function

    E.C. Titchmarsh, The theory of the Riemann zeta-function . Second edition. Edited and with a preface by D. R. Heath–Brown. The Clarendon Press, Oxford University Pre ss, New York, 1986. x+412 pp

    1986

  39. [47]

    A. B. Venkov, Spectral theory of automorphic functions and its applicati ons. Translated from the Russian by N. B. Lebedinskaya. Mathematics and its Applications (So viet Series), 51. Kluwer Academic Publishers Group, Dordrecht, 1990. xiv+176 pp

    1990

  40. [48]

    Xu, Nonvanishing of automorphic L-functions at special points

    Z. Xu, Nonvanishing of automorphic L-functions at special points . Acta Arith. 162 (2014), no. 4, 309–335. School of Mathematics and Statistics, Shandong University , Weihai, Weihai, Shandong 264209, China Email address : qfsun@sdu.edu.cn School of Mathematics, Shandong Univer...

    2014

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.