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

Zhi Qi

REVIEW 2 major objections 4 minor 34 references

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

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that at least one third of the special Hecke--Maass $L$-values $L(\tfrac12+it_f,f)$ with $t_f\le T$ are non-zero, with the proportion rising to one half under the Riemann hypothesis.

desk verdict A real result: first effective 33% non-vanishing for special Hecke–Maass L-values, with the main soft spot being the deferred Bessel lemmas that deserve a careful referee check. read the letter →

arxiv 2507.14566 v1 pith:EWTYYNT2 submitted 2025-07-19 math.NT

Zhi Qi This is my paper · ORCID

classification math.NT MSC 11M4111F72

keywords MaassformsL-functionsnon-vanishingKuznetsovtraceformulaspectralmomentsmollifierlow-lyingzerosdensitytheorem

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

The paper studies the non-vanishing of Hecke--Maass $L$-functions at the special point $s=\tfrac12+it_f$, the point attached to the Laplace eigenvalue of the cusp form. It proves that among forms with $t_f\le T$, at least one third of the values $L(\tfrac12+it_f,f)$ are non-zero, uniformly for even forms, odd forms, and the full orthonormal basis; on short intervals $|t_f-T|\le T^\mu$, the unconditional proportion is at least $\min\{1/3,(2\mu+1)/(2\mu+5)\}$. These are explicit constants, turning an earlier result that only gave an unspecified positive proportion into an effective one. For comparison, the central values $L(1/2,f)$ are known to be non-vanishing in at least 25\% of cases and vanish trivially on the odd basis, while the special values treated here escape that obstruction. Under the Riemann hypothesis the long-interval proportion rises to one half, with the curious caveat that RH does not improve the short-interval bound for $\mu\le 1/2$.

What carries the argument

The argument is carried by two kinds of spectral averages. First, the twisted first and second moments $C_\delta^1(m)$ and $C_\delta^2(m_1,m_2)$ over the even or odd Maass spectrum, weighted by the harmonic weight $\omega_f$ and a Gaussian in $t_f$. The Kuznetsov trace formula splits these averages into diagonal, Eisenstein, and off-diagonal terms; the off-diagonal terms are sums of Kloosterman sums against Bessel transforms, and the Bessel analysis in Section 4 (with full proofs deferred to the companion paper [Qi]) gives the asymptotics of those transforms, while Poisson summation and standard bounds for Kloosterman sums control the off-diagonal terms. The resulting asymptotic formulae, Theorems 5 and 6, supply explicit main terms: $C_\delta^1(m)=\Pi T/(\pi\sqrt\pi)\delta(m,1)+\cdots$ and $C_\delta^2(m_1,m_2)$ with main term $(\Pi T/(\pi\sqrt\pi r))\big((\log T/r+\gamma_\delta)\Sigma(\mathbf m)-2\bar\Sigma(\mathbf m)\big)$, where $r=m_1m_2/(m_1,m_2)^2$ and $\Sigma$ is a divisor sum. Second, a mollifier---a short Dirichlet polynomial whose coefficients are optimized by divisor inversion and the Prime Number Theorem---feeds these moments into Cauchy's inequality, producing the one-third lower bound; an unsmoothing lemma converts the smooth weight to sharp intervals. For the Riemann-hypothesis results, the same spectral machinery is applied to the one-level density at the special point through the explicit formula, and an extended density theorem with support $v(\mu)=\min\{3\mu,1+\mu\}$ is proved using variant Kloosterman sums and an identity that splits an ordinary Kloosterman sum into them.

What would settle it

A direct computation for a full spectrum up to large $T$ showing fewer than one third of $L(\tfrac12+it_f,f)$ non-zero would falsify Theorem 1; since that is currently out of reach, the sharpest available check is numerical verification of the Bessel integrals in Lemmas 4.1--4.3, for example the asymptotic (4.7), because the moment asymptotics and hence the one-third bound rest on them.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the special values $L(s_f,f)$ with $s_f=\tfrac12+it_f$ are non-vanishing for an explicit positive proportion of the spectrum, with the same constant on both parities: for $\delta=0,1$ and for the full basis $\mathcal B$, the liminf over $T$ of the proportion of $f$ with $t_f\le T$ and $L(s_f,f)\ne0$ is at least $1/3$. For short intervals the liminf is at least $\min\{1/3,(2\mu+1)/(2\mu+5)\}$, refining an earlier unexplicit result. Assuming the Riemann hypothesis for all $L(s,f)$ and Dirichlet $L$-functions, the long-interval proportion rises to at least $1/2$, and for $1/2<\mu<1$ the short-interval proportion rises to $\mu/(\mu+1)$; for small $\mu$ the conditional method does not beat the unconditional one, a feature the paper highlights. An addendum improves the conditional non-vanishing of central values as well, giving liminf greater than $9/16$ on the even basis and greater than $15/16$ on the odd basis.

Load-bearing premise

The load-bearing premise is that the technical estimates for Bessel-function integrals used in the moment computations are correct: Lemmas 4.1--4.3 are deferred to a companion paper, and the odd-integral extension is stated without full detail, so the one-third proportion stands or falls with that analysis.

Editorial extensions

If this is right

For each parity $\delta=0,1$ and for the full basis $\mathcal B$, at least one third of the special values $L(\tfrac12+it_f,f)$ with $t_f\le T$ do not vanish as $T\to\infty$.
On windows $|t_f-T|\le T^\mu$, $0<\mu<1$, the unconditional non-vanishing proportion is at least $\min\{1/3,(2\mu+1)/(2\mu+5)\}$, making the earlier unspecified proportion effective.
Under the Riemann hypothesis for $L(s,f)$ and Dirichlet $L$-functions, the long-interval proportion is at least $1/2$, and for $1/2<\mu<1$ the short-interval proportion is at least $\mu/(\mu+1)$; for $\mu\le 1/2$ the conditional bound is weaker than the unconditional $1/3$.
The asymptotic formulae for the twisted first and second moments give the leading-order distribution of $L(\tfrac12+it_f,f)$ at the special point, which can serve as a quantitative baseline for further statistics of these values.
Because the special values do not vanish trivially on the odd basis, the $1/3$ lower bound applies equally to both parity classes, unlike central values.

Reading between the lines

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

The one-third constant is a lower bound supplied by the method, not a prediction of the true proportion; the conditional jump to one half suggests that improving the unconditional off-diagonal and Bessel control is the natural route to larger constants.
The same moment machinery, with the deferred Bessel estimates in place, could plausibly be applied to other families evaluated at their spectral points, such as Rankin--Selberg values connected to cusp-form deformation questions, to replace unspecified proportions with explicit ones.
A numerical check on moderately large $T$ comparing the computed non-vanishing ratio of $L(\tfrac12+it_f,f)$ with $1/3$, $1/2$, and the short-interval constants would be a testable extension, and a ratio systematically above $1/3$ would support the view that the true proportion is larger.
The saturation of the unconditional bound for $\mu\le 1/2$ indicates that on very short intervals the bottleneck is the diagonal--mollifier balance rather than the zero-density range, so subconvexity or refined mollifier choices may be the right extension target.

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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

Punchline: this is a real result. The author proves the first effective non-vanishing proportion (33%) for the family L(1/2+it_f, f), raises it to 50% under GRH, and adds short-interval versions. Xu only had an ineffective positive proportion for this family, and the twisted moment asymptotics plus the extended density theorem are new. The work is technically substantial and appears to be carried out carefully.

What I like: the moment calculations are organized, the error terms are precisely stated, and the mollifier optimization is standard but executed correctly. The paper is honest about where hypotheses enter, and the conditional results are cleanly separated from the unconditional ones.

The soft spot is real but manageable: the load-bearing Bessel analysis (Lemmas 4.1–4.3) is deferred to the companion paper [Qi], and the extension to the H^- integral is asserted as “literally the same proofs.” The stress-test note is right that the H^- phase f_-(r;v,w) = v e^r − w e^{-r} differs from the H^+ phase, so the stationary-phase analysis is not literally identical. That does not mean the claim is wrong—the stationary-phase condition is analogous and the resulting truncation is plausible—but it is a verifiability gap. A referee should check it, and the author should either include the proofs or pin the companion paper firmly into the record. The paper does display the actual phases and integral representations in (4.7)–(4.8), so a referee has a concrete starting point.

The §10.4 “max”/“min” typo is real; the final theorem and (10.28) use the correct “min,” so it is harmless.

I do not see circularity: the self-citations supply auxiliary transforms and Weyl laws, not the non-vanishing result. The citation pattern is appropriate.

My verdict: this deserves a serious referee. The main theorem is a genuine advance, and the proof structure is sound modulo the deferred Bessel lemmas. I would recommend publication after the deferred analysis is made self-contained or the companion paper is firmly integrated, and after the typo is fixed. It is a good candidate for a reading group, though the density-theorem part is heavier going. Yes, accept for peer review.

Referee Report

2 major / 4 minor

Summary. The paper proves nonvanishing results for Hecke-Maass L-functions at the special point s_f = 1/2 + it_f. The main unconditional theorem (Theorem 1) states that for each parity class B_delta, at least one third of the values L(s_f, f) with t_f <= T are nonzero as T grows, and the same holds on the full basis B. Theorem 2 gives a short-interval version with proportion min{1/3, (2mu+1)/(2mu+5)}. Under the Riemann hypothesis for the relevant L-functions and Dirichlet L-functions, Theorems 3 and 4 raise the long-interval proportion to 1/2 and give mu/(mu+1) for mu > 1/2. The proof combines the Kuznetsov formula, approximate functional equations, Bessel-integral analysis, a mollifier method, and an extended density theorem for the 1-level density of L(s_f, f). The paper is technically dense and the main line of argument is plausible, but several load-bearing estimates are deferred or misstated and need correction.

Significance. If the central claims are correct, this is a substantial advance: it gives the first explicit nonvanishing proportion above the 25% known for central values, does so separately in the even and odd families, and includes effective short-interval refinements. The method is a careful combination of standard tools, and the paper is unusually explicit about the constants and error terms. The density-theorem part identifies the unitary symmetry of the family and is of independent interest. The main caveat is that the unconditional 33% result depends on Bessel-integral estimates whose H^- variant is deferred to a companion paper; this is a genuine load-bearing point. The paper does not provide machine-checked proofs or code, but the analytic structure is transparent enough that verification is feasible.

major comments (2)

[Section 4, Lemmas 4.1-4.3 and Corollary 4.1] The H^- extension of the Bessel-integral analysis is asserted to follow "by literally the same proofs" from the companion paper [Qi], but the phase in Lemma 4.2 is f_-(r;v,w)=v e^r - w e^{-r}, obtained through the I-Bessel identity (2.8), and the stationary-phase analysis is not literally identical to the H^+ case. Since Corollary 4.1 supplies the c-truncation used in Lemmas 8.2 and 9.3, and hence the error term in Theorem 6, the 1/3 nonvanishing constant depends directly on this deferred analysis. The paper should either include the H^- proofs or give a precise statement of the modifications needed and confirm that the companion paper contains them.
[Section 9.2, Lemma 9.2, and Theorem 6] The error term in Lemma 9.2 and Theorem 6 is stated as O(T^epsilon( Pi sqrt(T) + T sqrt(r) + Pi^3/(T sqrt(r)) )), but the derivation leading to (9.12) yields an error of order T^{1+epsilon}/sqrt(r) from the diagonal/off-diagonal split, together with Pi T^{1/2+epsilon} and Pi^3 T^epsilon/(T sqrt(r)). With the printed T sqrt(r), the contribution to the mollified second moment in Section 10.3 would be of size T times a sum of 1/gcd(m1,m2), which is about T M^2; this is incompatible with the constraints in Lemma 10.2 and would block the nonvanishing proportion. The text in Section 10.3 appears to use the correct T/sqrt(r) form, so the statements in Lemma 9.2 and Theorem 6 should be corrected and all later uses checked.

minor comments (4)

[Section 10.4, equations (10.23) and (10.24)] The displayed optimization uses "max" where the argument and the final statements (10.28) and Theorem 2 use "min". The printed max would claim a proportion exceeding 1/3 in the range mu > 1/2, which is not what the method proves.
[Section 15.2, last sentence] The phrase "Lemma (15.1)" should read "Lemma 15.1".
[Section 10.4, removal of the harmonic weight] The passage from the weighted lower bound to the unweighted statements in Theorems 1 and 2 is delegated to the Kowalski-Michel method and a citation to [BHS]. Since Theorem 1 is an unweighted density statement, it would be helpful to spell out the specific lemma or adaptation being used.
[Section 4, Remark 4.1] The even case requires a split treatment with two different choices of Re(v) and regularity parameters. This is plausible, but a short explanation of why the two regimes overlap and cover all relevant u would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 1/3 non-vanishing bound follows from independently computed twisted moments and an optimized mollifier; deferred Bessel lemmas are a completeness gap, not a circular step.

full rationale

The proof of Theorem 1 is a standard mollifier-moment argument: the first and second twisted moments are evaluated by the Kuznetsov trace formula and approximate functional equations (Theorems 5 and 6), and the mollifier coefficients are chosen by Cauchy's inequality to maximize the ratio M1^2/M2. No parameter is fitted to the non-vanishing statistic, and no equation defining the target quantity is reused as a conclusion. The self-citations to [Qi], [LQ1], and [LQ2] supply auxiliary analytic facts (Bessel transforms, weighted Weyl law, unsmoothing) whose assumptions are stated in terms of test functions and spectral parameters, not in terms of the non-vanishing proportion being proved. In particular, Corollary 4.1 depends on Lemmas 4.1-4.3 whose proofs are deferred to the companion paper [Qi]; this is a completeness/deferral risk, not a circular reduction, because the companion results are independent Bessel-integral estimates and do not presuppose the 33% conclusion. The 'max' in (10.23) is a typographical slip corrected by 'min' in Theorem 2 and (10.28). No circularity score above 0 is warranted.

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

The unconditional theorems rest on standard analytic number theory (Kuznetsov formula, Weil bound, approximate functional equations) plus Bessel estimates taken from a companion preprint by the same author. The conditional theorems add the Riemann hypothesis for Dirichlet L-functions and for the family L(s,f). No empirical fitting or invented entities appear.

assumptions (4)

standard math Kuznetsov trace formula for even and odd Maass cusp forms on SL_2(Z), equation (2.9), is valid.
This is the main spectral tool used to evaluate the twisted first and second moments in Sections 8 and 9. It is a proved theorem in the literature, cited as [CI, §3].
standard math Kim-Sarnak bound |α_f(p)|, |β_f(p)| ≤ p^{7/64} in equation (3.7) holds for all primes p.
Used in Section 14 to discard higher prime powers in the explicit formula for the 1-level density. It is a deep proven theorem, cited as [Kim].
ad hoc to paper The Bessel integral estimates in Lemmas 4.1-4.3, including the H^- case, are valid as stated, with proofs deferred to the companion paper [Qi, arXiv:2506.08546].
These estimates control the off-diagonal terms in the twisted moments. The paper says the H^- case follows from the same proofs as in [Qi], but the full arguments are not repeated here, making this an assumed auxiliary result.
domain assumption The Riemann hypothesis for Dirichlet L-functions and for the family L(s,f) holds, for Theorems 3, 4, 9, and 10.
The conditional results explicitly assume RH for Dirichlet L-functions and, in Theorems 3 and 4, also for every L(s,f) with f in B_δ. This is an unproved conjecture, stated as a hypothesis in the paper.

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Pith. "Pith review of On the Effective Non-vanishing of Hecke--Maass $L$-functions at Special Points." pith.science (2026). https://pith.science/paper/EWTYYNT2

@misc{pith2026250714566,
  author       = {Pith},
  title        = {Pith review of: On the Effective Non-vanishing of Hecke--Maass $L$-functions at Special Points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EWTYYNT2}},
  note         = {Machine review of arXiv:2507.14566}
}

abstract

In this paper, we consider the non-vanishing problem for the family of special Hecke--Maass $L$-values $ L (1/2+it_f, f) $ with $f (z)$ in an orthonormal basis of (even or odd) Hecke--Maass cusp forms of Laplace eigenvalue $1/4 + t_f^2$ ($t_f > 0$). We prove that 33% of $L (1/2+it_f, f)$ for $ t_f \leqslant T$ do not vanish as $T \rightarrow \infty$. For comparison, it is known that the non-vanishing proportion is at least 25% for the central $L$-values $L (1/2, f)$. Further, 33% may be raised to 50% conditionally on the generalized Riemann hypothesis. Moreover, we prove non-vanishing results on short intervals $|t_f-T| \leqslant T^{\mu}$ for any $0 < \mu < 1$. However, it is a curious case that the Riemann hypothesis does not yield better result for small $0 < \mu \leqslant 1/2$.

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Works this paper leans on

34 extracted references · 33 canonical work pages

[1]

Alpoge and S

L. Alpoge and S. J. Miller. Low-lying zeros of M aass form L -functions. Int. Math. Res. Not. IMRN , (10):2678--2701, 2015

work page 2015
[2]

Balkanova, B

O. Balkanova, B. Huang, and A. S\" o dergren. Non-vanishing of M aass form L -functions at the central point. Proc. Amer. Math. Soc. , 149(2):509--523, 2021

work page 2021
[3]

Bohr and E

H. Bohr and E. Landau. Sur les z\'eros de la fonction (s) de R iemann. C. R. Acad. Sci. Paris , 158:106--110, 1914

work page 1914
[4]

V. Blomer. Subconvexity for twisted L -functions on GL (3) . Amer. J. Math. , 134(5):1385--1421, 2012

work page 2012
[5]

Bourgain

J. Bourgain. Decoupling, exponential sums and the R iemann zeta function. J. Amer. Math. Soc. , 30(1):205--224, 2017

work page 2017
[6]

Bourgain and N

J. Bourgain and N. Watt. Decoupling for perturbed cones and the mean square of | ( 12+it)| . Int. Math. Res. Not. IMRN , (17):5219--5296, 2018

work page 2018
[7]

J. B. Conrey and H. Iwaniec. The cubic moment of central values of automorphic L -functions. Ann. of Math. (2) , 151(3):1175--1216, 2000

2000
[8]

Deshouillers and H

J.-M. Deshouillers and H. Iwaniec. The nonvanishing of R ankin- S elberg zeta-functions at special points. The S elberg T race F ormula and R elated T opics ( B runswick, M aine, 1984) , Contemp. Math., Vol. 53 , 51--95. Amer. Math. Soc., Providence, RI, 1986

work page 1984

Show all 34 references

[9]

Ivi\' c and M

A. Ivi\' c and M. Jutila. On the moments of H ecke series at central points. II . Funct. Approx. Comment. Math. , 31:93--108, 2003

2003
[10]

Iwaniec and E

H. Iwaniec and E. Kowalski. Analytic N umber T heory , American Mathematical Society Colloquium Publications, Vol. 53 . American Mathematical Society, Providence, RI, 2004

2004
[11]

Iwaniec and X

H. Iwaniec and X. Li. The orthogonality of H ecke eigenvalues. Compos. Math. , 143(3):541--565, 2007

2007
[12]

Iwaniec, W

H. Iwaniec, W. Luo, and P. Sarnak. Low lying zeros of families of L -functions. Inst. Hautes \'Etudes Sci. Publ. Math. , (91):55--131 (2001), 2000

2001
[13]

Iwaniec and P

H. Iwaniec and P. Sarnak. The non-vanishing of central values of automorphic L -functions and L andau- S iegel zeros. Israel J. Math. , 120(part A):155--177, 2000

2000
[14]

H. H. Kim. Functoriality for the exterior square of GL _4 and the symmetric fourth of GL _2 . J. Amer. Math. Soc. , 16(1):139--183, 2003. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak

2003
[15]

Kowalski and P

E. Kowalski and P. Michel. The analytic rank of J_0(q) and zeros of automorphic L -functions. Duke Math. J. , 100(3):503--542, 1999

1999
[16]

Kowalski and P

E. Kowalski and P. Michel. A lower bound for the rank of J_0(q) . Acta Arith. , 94(4):303--343, 2000

2000
[17]

N. M. Katz and P. Sarnak. Random M atrices, F robenius E igenvalues, and M onodromy , American Mathematical Society Colloquium Publications . Vol. 45. American Mathematical Society, Providence, RI, 1999

1999
[18]

X. Li. Bounds for GL (3) GL (2) L -functions and GL (3) L -functions. Ann. of Math. (2) , 173 (1):301--336, 2011

2011
[19]

X. Li. A weighted W eyl law for the modular surface. Int. J. Number Theory , 7(1):241--248, 2011

2011
[20]

S.-C. Liu. and Z. Qi. Low-lying zeros of L -functions for M aass forms over imaginary quadratic fields. Mathematika , 66(3):777--805, 2020

2020
[21]

Liu and Z

S.-C. Liu and Z. Qi. Moments of central L -values for M aass forms over imaginary quadratic fields. Trans. Amer. Math. Soc. , 375(5):3381--3410, 2022

2022
[22]

W. Luo. On the nonvanishing of R ankin- S elberg L -functions. Duke Math. J. , 69(2):411--425, 1993

1993
[23]

W. Luo. The spectral mean value for linear forms in twisted coefficients of cusp forms. Acta Arith. , 70(4):377--391, 1995

1995
[24]

W. Luo. Nonvanishing of L -values and the W eyl law. Ann. of Math. (2) , 154(2):477--502, 2001

2001
[25]

W. Luo. Refined asymptotics for second spectral moment of R ankin- S elberg L -functions at the special points. Int. Math. Res. Not. IMRN , (23):5457--5483, 2012

2012
[26]

Magnus, F

W. Magnus, F. Oberhettinger, and R. P. Soni. Formulas and T heorems for the S pecial F unctions of M athematical P hysics . 3rd enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52. Springer-Verlag New York, Inc., New York, 1966

1966
[27]

R. S. Phillips and P. Sarnak. On cusp forms for co-finite subgroups of PSL (2, R ) . Invent. Math. , 80(2):339--364, 1985

1985
[28]

Z. Qi. O n the effective non-vanishing of R ankin-- S elberg L -functions at special points. arXiv: 2506.08546 , 2025

2025 arXiv
[29]

Rudnick and P

Z. Rudnick and P. Sarnak. Zeros of principal L -functions and random matrix theory. Duke Math. J. , 81(2):269--322, 1996

1996
[30]

A. Selberg. On the zeros of R iemann's zeta-function. Collected Papers. V ol. I , Contemp. Math., Vol. 53 , 85--141. Springer-Verlag, Berlin, 1989

1989
[31]

E. C. Titchmarsh. The T heory of the R iemann Z eta- F unction . The Clarendon Press, Oxford University Press, New York, 2nd edition, 1986. Edited and with a preface by D. R. Heath-Brown

1986
[32]

G. N. Watson. A T reatise on the T heory of B essel F unctions . Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944

1944
[33]

Z. Xu. Nonvanishing of automorphic L -functions at special points. Acta Arith. , 162(4):309--335, 2014

2014
[34]

M. P. Young. Weyl-type hybrid subconvexity bounds for twisted L -functions and H eegner points on shrinking sets. J. Eur. Math. Soc. (JEMS) , 19(5):1545--1576, 2017

2017

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Alpoge and S

L. Alpoge and S. J. Miller. Low-lying zeros of M aass form L -functions. Int. Math. Res. Not. IMRN , (10):2678--2701, 2015

work page 2015

[2] [2]

Balkanova, B

O. Balkanova, B. Huang, and A. S\" o dergren. Non-vanishing of M aass form L -functions at the central point. Proc. Amer. Math. Soc. , 149(2):509--523, 2021

work page 2021

[3] [3]

Bohr and E

H. Bohr and E. Landau. Sur les z\'eros de la fonction (s) de R iemann. C. R. Acad. Sci. Paris , 158:106--110, 1914

work page 1914

[4] [4]

V. Blomer. Subconvexity for twisted L -functions on GL (3) . Amer. J. Math. , 134(5):1385--1421, 2012

work page 2012

[5] [5]

Bourgain

J. Bourgain. Decoupling, exponential sums and the R iemann zeta function. J. Amer. Math. Soc. , 30(1):205--224, 2017

work page 2017

[6] [6]

Bourgain and N

J. Bourgain and N. Watt. Decoupling for perturbed cones and the mean square of | ( 12+it)| . Int. Math. Res. Not. IMRN , (17):5219--5296, 2018

work page 2018

[7] [7]

J. B. Conrey and H. Iwaniec. The cubic moment of central values of automorphic L -functions. Ann. of Math. (2) , 151(3):1175--1216, 2000

2000

[8] [8]

Deshouillers and H

J.-M. Deshouillers and H. Iwaniec. The nonvanishing of R ankin- S elberg zeta-functions at special points. The S elberg T race F ormula and R elated T opics ( B runswick, M aine, 1984) , Contemp. Math., Vol. 53 , 51--95. Amer. Math. Soc., Providence, RI, 1986

work page 1984