A zero density estimate for Dedekind zeta functions

Asif Zaman; Jesse Thorner

REVIEW 2 major objections 3 minor 23 references

A zero density estimate for Dedekind zeta functions

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read This paper proves an unconditional zero density estimate for the Dedekind zeta functions of Galois extensions with any fixed finite Galois group, removing the need for the strong Artin conjecture in recent Chebotarev and class-group…

desk verdict Genuinely new unconditional zero-density setup for nonabelian Galois families, but the written proof has a concrete exponent gap in Cor. 6.2 that leaves Thm 1.1 unproved as stated. read the letter →

arxiv 1909.01338 v2 pith:NQYTRIGC submitted 2019-09-03 math.NT

Jesse Thorner , Asif Zaman This is my paper · ORCID

classification math.NT MSC 11M4111R4211N3511R2911R45

keywords zerodensityestimatesDedekindzetafunctionsArtinL-functionslargesieveinequalityChebotarevtheoremclassgrouptorsionGaloisextensionsstrongconjecture

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 proves the first zero density estimate for Dedekind zeta functions of Galois extensions with a fixed finite Galois group $G$ that does not assume unproven progress toward the strong Artin conjecture. It shows that, over fields $K$ with discriminant at most $Q$, the number $N_K(\sigma,T)$ of zeros of $\zeta_K(s)/\zeta(s)$ in the region $\beta\ge \sigma$, $|\gamma|\le T$, satisfies $\sum_{K\in \mathcal{F}(Q)} N_K(\sigma,T) \ll_m m_{\mathcal{F}}(Q)(QT)^{107m^3(1-\sigma)}(\log QT)^{2m^2}$, where $m+1=|G|$ and $m_{\mathcal{F}}(Q)$ counts how many fields in the family share a nontrivial subfield. This makes it possible to average the error term in the Chebotarev density theorem, prove subconvexity for $\zeta_k(1/2)$, and bound $\ell$-torsion in class groups across such families, all without the strong Artin conjecture. The price is a hypothesis on how many fields in the family share a common subfield, which is known to hold for simple Galois groups such as $A_n$.

What carries the argument

The engine is the large sieve inequality of Theorem 6.1. For each $K$, write $L(s,\chi_K)=\zeta_K(s)/\zeta(s)=\sum_n a_K(n)n^{-s}$. The coefficients are encoded through Schur polynomials at the local roots, and the Cauchy identity rewrites the correlation sum $\sum_{(n,D_KD_{K'})=1}a_{K\times K'}(n)\phi(T\log n\,/\,x)$ as a contour integral of the Artin $L$-function $L(s,\chi_K\otimes\chi_{K'})$. When $K\cap K'=\mathbb{Q}$, the holomorphy and conductor bound of Proposition 4.3 let the contour shift to $\mathrm{Re}(s)=1/2$, yielding the saving $\sqrt{x}\,Q^{m/2+\varepsilon}T^{m^2}$; when the fields intersect, a trivial divisor-counting bound loses only the factor $m_{\mathcal{F}}(Q)$. A standard mean-value identity from the large sieve literature converts the resulting short-interval bound into an $L^2$-averaged bound over primes, and the zero-detection lemmas of Section 7 convert that bound into the zero count $N_K(\sigma,T)$.

What would settle it

For a small group such as $G=S_3$ ($m=5$), enumerate all $S_3$-Galois fields with $D_K\le Q$, compute the zeros of $\zeta_K/\zeta$ up to height $T$ with numerical routines, and compare $\sum_{K\in\mathcal{F}(Q)}N_K(\sigma,T)$ with $m_{\mathcal{F}}(Q)(QT)^{107m^3(1-\sigma)}(\log QT)^{2m^2}$; if the ratio is unbounded for a sequence of $Q,T$, Theorem 1.1 is false. More directly, exhibiting linearly disjoint Galois $K,K'$ for which $\zeta_{KK'}\zeta/(\zeta_K\zeta_{K'})$ has a pole inside the critical strip would break Lemma 5.1 and remove the diagonal control of the large sieve.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for any nontrivial finite group $G$ of order $m+1$, the sum over Galois extensions $K/\mathbb{Q}$ with $\mathrm{Gal}(K/\mathbb{Q})\cong G$ and $D_K\le Q$ of the zero count $N_K(\sigma,T)$ for $\zeta_K(s)/\zeta(s)$ is $\ll_m m_{\mathcal{F}}(Q)(QT)^{107m^3(1-\sigma)}(\log QT)^{2m^2}$ for $1/2\le \sigma\le 1$. The exponent $107m^3(1-\sigma)$ is the essential feature: near $\sigma=1$, very few zeros occur, which is precisely what average prime counting requires. The novelty is that the averaging does not require each $\zeta_K/\zeta$ to be automorphic; instead, the proof uses a large sieve inequality for the Dirichlet coefficients of these Artin $L$-functions, whose diagonal part is controlled by the holomorphy, quoted as Proposition 4.3, of the tensor-product Artin $L$-function $L(s,\chi_K\otimes\chi_{K'})$ for linearly disjoint fields. From the zero density theorem, the paper derives a positive level of distribution for Chebotarev primes, subconvexity of Dedekind zeta functions, and nontrivial class-group torsion bounds, all under the single hypothesis $m_{\mathcal{F}}(Q)\ll_{m,\varepsilon}Q^{-2\varepsilon}|\mathcal{F}(Q)|$.

Load-bearing premise

The load-bearing external input is the theorem, quoted as Proposition 4.3, that for linearly disjoint Galois extensions $K,K'/\mathbb{Q}$ the tensor-product Artin $L$-function $L(s,\chi_K\otimes\chi_{K'})$ is entire with conductor dividing $D_K^{m'}D_{K'}^m$; if that holomorphy were unavailable, the diagonal estimate in the large sieve (Lemma 5.1) and hence the whole Theorem 1.1 would collapse.

Editorial extensions

If this is right

With condition (2.6), the averaged Chebotarev error over all conjugacy classes is $O(x/(\log x)^A)$ for $\log Q$ up to $x^{\varepsilon/(109m^3)}$, a positive level of distribution without GRH or the strong Artin conjecture.
For the completely split conjugacy class, the average error is $O(x^{1-\delta})$, a stronger saving on the primes that split completely.
All but $O(Q^{-\varepsilon}|\mathcal{F}(Q)|)$ fields have $|\zeta_k(1/2)|\ll D_k^{1/4-\delta/109}$ for every subfield $k$ whose Galois closure is in the family, giving subconvexity on average.
All but $O(Q^{-\varepsilon}|\mathcal{F}(Q)|)$ fields have $|\mathrm{Cl}_k[\ell]|\ll D_k^{1/2-1/(2\ell([k:Q]-1))+\eta}$, a nontrivial bound for $\ell$-torsion in class groups.
For $G=A_n$, $n\ge 5$, the hypotheses are unconditional, giving the first unconditional instances of these three conclusions for infinitely many unsolvable extensions of arbitrarily large degree.

Reading between the lines

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

Editorial inference beyond the paper: the same large sieve should apply to any family of Artin representations for which the tensor-product $L$-functions are known to be entire from a source other than automorphy; the method is not logically tied to the specific quotient $\zeta_K/\zeta$.
Editorial inference beyond the paper: the exponent $107m^3$ is presumably far from sharp; any sharper conductor estimate for $L(s,\chi_K\otimes\chi_{K'})$ would lower the exponent and improve the level-of-distribution exponent $\delta=\varepsilon/(109m^3)$.
Editorial inference beyond the paper: condition (2.6) turns into a discriminant multiplicity problem for groups with a unique nontrivial normal subgroup; the paper's treatment of $S_n$ shows that resolving enough of the discriminant multiplicity conjecture would make the $S_n$ case unconditional.
Editorial inference beyond the paper: Theorem 1.1 also provides, for most fields in such families, a zero-free region of width roughly $1/\log Q$ near the edge of the critical strip, which is the natural unconditional proxy for GRH in these families and could serve as the input to single-field effective Chebotarev estimates.

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.

Desk Editor's Note

read the letter

Bottom line: this is a real theorem, but the written proof has a gap at a badly chosen exponent. The main new input is a large sieve for coefficients of ζ_K/ζ without assuming automorphy, using Brauer's holomorphy for tensor L-functions of linearly disjoint fields. That is genuinely new, and the zero-detection setup from Soundararajan–Thorner fits cleanly. The applications to Chebotarev, subconvexity, and torsion are honestly stated, with the S_n case flagged as conditional on the discriminant/biquadratic counting condition (2.7). I don't see circularity or fitted constants; the self-citations are for methodology, not for the central claim.

The soft spot is concrete. Corollary 6.2 as stated requires y ≫_m (QT)^{10^8(m+1)}; its proof, however, requires y ≫_m Q^{10^8(m+1)}T^{2m^2}. More importantly, in the proof of Theorem 1.1 they apply Corollary 6.2 with y = N0 = exp(M/(300η)), where M = 2·10^5 m^3 η log(QT), so N0 = (QT)^{(2/3)·10^3 m^3}. That is about (QT)^{6.7·10^2 m^3}, not (QT)^{10^8(m+1)}. The paper's line saying that (2·10^5)/300 ≥ 10^8(m+1) is off by roughly five orders of magnitude. Thus the averaging step that bridges the large sieve and the zero count has no admissible y under the stated hypotheses, and Theorem 1.1 is unproved as written. This is not a cosmetic typo: the natural fix, choosing M much larger, would blow up the final exponent unless other parts are reworked.

Despite that, I think the proof strategy is sound and the theorem is likely true with larger exponents. The Brauer tensor input is external but standard; the Chebotarev machinery in Sections 8–10 is careful and mostly self-contained. This is the kind of paper that deserves referee time, but the authors need to repair Corollary 6.2 and the application in §7 before the main theorem is complete. I would not cite Theorem 1.1 as it stands, but I would read a revised version.

Recommendation: send to peer review rather than desk reject. The result is important enough and the framework is novel enough that a serious referee should look at it, with the explicit task of checking whether the y-range can be repaired without changing the final exponent.

Referee Report

2 major / 3 minor

Summary. The paper proves a zero density estimate for Dedekind zeta functions associated to Galois extensions K/Q with a fixed finite Galois group G, counting zeros of the quotient ζ_K(s)/ζ(s) in rectangles of height T and distance 1−σ from the critical line. The result is unconditional and does not assume the strong Artin conjecture, which previous work required. The proof combines Brauer's holomorphy theorem for tensor products of Artin characters, a large sieve inequality for Dedekind zeta coefficients, and the zero-detection method of Soundararajan–Thorner. Applications are given to the average error in the Chebotarev density theorem, subconvexity of Dedekind zeta functions, and ℓ-torsion in class groups.

Significance. If the proof is completed, the main theorem is a significant advance: it appears to be the first zero density estimate for families of Dedekind zeta functions with arbitrary fixed Galois group without assuming strong Artin, and it yields new unconditional applications to Chebotarev, subconvexity, and torsion. The paper is well-structured, uses external theorems such as Brauer's result appropriately, and does not rely circularly on the strong Artin conjecture or on the conclusions it aims to derive. The large sieve framework and the zero-detection argument are coherent in outline, and the claimed applications are clearly stated.

major comments (2)

[§6, Corollary 6.2 and §7, proof of Theorem 1.1] The y-range in Corollary 6.2 is inconsistent with both its proof and its application. The corollary as stated requires y ≫_m (QT)^{10^8(m+1)}, while the proof obtains the conclusion only under y ≫_m Q^{10^8(m+1)} T^{2m^2}. These two conditions differ in the exponent of T, and neither condition is satisfied by the value y = N0 used in §7. In the proof of Theorem 1.1, N0 = exp(M/(300η)) with M = 2·10^5 m^3 η log(QT), so N0 = (QT)^{(2·10^5 m^3)/300} = (QT)^{(2000/3)m^3}. For m = 1, this is (QT)^{666.7}, which is far smaller than the lower bound (QT)^{2·10^8} required by the stated corollary and also smaller than Q^{2·10^8} T^2 required by the proof of the corollary when T is small. The assertion in §7 that the application is valid because (2·10^5 m^3)/300 ≥ 10^8(m+1) for m ≥ 1 is numerically false. Consequently the averaging step over K in (7.8) is not justified as written. This is a load-bearing gap in the proof of Theorem 1.1: either Corollary 6.2 must be proved with a y-condition that is actually satisfied by N0 (e.g., by improving the large sieve or by choosing a smaller power of Q), or the proof must choose a larger N0 and recompute the resulting exponent in Theorem 1.1, which will likely weaken the final zero density estimate.
[§8, Proposition 8.7] The dyadic construction of a zero-free region does not establish the claimed region. The proof applies Theorem 1.1 with T = T_j = e^j − 3 and σ = σ_j = 1 − 20δ log Q / (log Q + log(T_j + 3)). This only shows that, for non-exceptional fields, there are no zeros with β ≥ σ_j and |γ| ≤ T_j. However, the claimed zero-free region has boundary α(t) = 1 − 20δ log Q / (log Q + log(t + 3)) for a zero of height t. Since α(t) < σ_j for t < T_j, a zero with β > α(t) but β < σ_j is not excluded by the argument. Thus the conclusion that Δ_{K/Q}(t) ≥ α(t) for all 3 ≤ t ≤ exp(Q^{ε/2}) does not follow. To obtain the claimed region, one would need to apply the density estimate with σ = α(T_{j−1}) and T = T_j (or use a similar pointwise argument), and the resulting exceptional-set estimate would need to be rechecked. This affects the proof of Theorem 2.1 and Corollary 2.2, which rely on Proposition 8.7.

minor comments (3)

[Throughout, especially §6 and §7] Several exponents are difficult to read in the plain-text version, such as Q^{54(m+1)} and Q^{108(m+1)}; these should be typeset unambiguously, and the numerical comparison in §7 should be corrected to match the actual inequalities used in the proof of Corollary 6.2.
[§2.1 and references] The paper relies heavily on the authors' earlier works [4], [21], and [23] for the zero-detection lemmas and the Chebotarev refinement. It would help the reader if Section 2.1 explicitly listed which results are quoted from those papers and which are new, particularly because the proof of Lemma 7.3 and the proof of Proposition 8.7 are only sketches that send the reader to [21] and [23].
[§5, Lemma 5.1] In the contour-integral proof of Lemma 5.1, the product over ramified primes is controlled by the bound ∏_{p|D_KD_{K'}} (1+p^{-1/2})^{m^2} ≪_ε Q^ε; this is correct, but the role of the conductor bound from Proposition 4.3 in the convexity estimate could be stated more explicitly for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 rests on an external Brauer holomorphy theorem, a self-contained large sieve, and general lemmas from prior published work that are applied, not assumed.

full rationale

The derivation of Theorem 1.1 does not reduce to its own inputs. The central new mechanism is the large sieve inequality Theorem 6.1, built from the coefficient identities (5.3) and (5.4) and the estimates Lemmas 5.1 and 5.2; Lemma 5.1 uses Proposition 4.3, which is Brauer's external theorem [3] on the holomorphy of L(s,chi_K tensor chi_{K'}), not an unproven strong Artin conjecture. Section 7 then applies the zero-detection lemmas from [21] after verifying L(s,chi_K) lies in the class S(m). These are transfer statements with stated hypotheses that do not include Theorem 1.1, so citing [21] and [23] (both by overlapping authors) is methodology, not circular evidence. The factor m_F(Q) is an input defined by field intersections in (1.1), and no fitted parameter is later relabeled as a prediction. The y-range mismatch flagged in Corollary 6.2 is a correctness/incompleteness concern about the application of the corollary, not a definitional reduction of the conclusion to the hypotheses, and therefore does not affect this circularity score.

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

The central theorem rests on standard analytic number theory: Artin L-function theory, Brauer's holomorphy results, convexity bounds, Stark's zero-free region, and Ellenberg-Venkatesh field counting. The applications add family hypotheses that are stated explicitly. There are no fitted parameters and no invented objects such as new particles, forces, or conserved quantities.

assumptions (8)

standard math Artin L-functions L(s,χ) have meromorphic continuation and a functional equation.
Invoked in Section 3 to define Λ(s,χ) and used throughout the zero-detection arguments.
standard math Aramata-Brauer theorem: ζ_K(s)/ζ(s) is entire for Galois K/Q.
Theorem 4.1; this puts L(s,χK) into the framework of [21] and justifies the Hadamard product and zero-counting estimates.
standard math Brauer's theorem: L(s,χK⊗χK') is entire for linearly disjoint Galois K and K'.
Proposition 4.3; the key analytic input for Lemma 5.1 and for the large sieve, since the tensor L-function is not known to be automorphic.
standard math Convexity bound |L(1/2+it,χK⊗χK')| ≪_m Q^{m/2}(2+|t|)^{m^2/4}.
Used in Lemma 5.1 and cited to Heath-Brown [9].
standard math Stark/Lagarias-Odlyzko zero-free region for Dedekind zeta functions, including the exceptional-zero bound.
Theorem 8.5 underlies Lemma 8.6 and Proposition 8.7.
standard math Counting bound #F(Q) ≪_m Q^{52(m+1)} from Ellenberg and Venkatesh.
Used at the end of the proof of Theorem 6.1 to control the number of fields in F(Q).
domain assumption Family condition (2.6): mF(Q) ≪_{m,ε} Q^{-2ε} #F(Q).
A hypothesis of Theorem 2.1; for A_n it is verified via simplicity and field counts, while for S_n it reduces to the unproved condition (2.7).
domain assumption For G=S_n, the maximum number of fields sharing a fixed quadratic resolvent is bounded by D^{1/2+1/n-2ε} (condition (2.7)).
Corollary 2.2(b) is explicitly conditional on this unproved discriminant-related bound.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A zero density estimate for Dedekind zeta functions." pith.science (2026). https://pith.science/paper/NQYTRIGC

@misc{pith2026190901338,
  author       = {Pith},
  title        = {Pith review of: A zero density estimate for Dedekind zeta functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NQYTRIGC}},
  note         = {Machine review of arXiv:1909.01338}
}

abstract

Given a nontrivial finite group $G$, we prove the first zero density estimate for families of Dedekind zeta functions associated to Galois extensions $K/\mathbb{Q}$ with $\mathrm{Gal}(K/\mathbb{Q})\cong G$ that does not rely on unproven progress towards the strong form of Artin's conjecture. We use this to remove the hypothesis of the strong Artin conjecture from the work of Pierce, Turnage-Butterbaugh, and Wood on the average error in the Chebotarev density theorem and $\ell$-torsion in ideal class groups.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Squarefree values of polynomial discriminants I

M. Bhargava, A. Shankar, and X. Wang. Squarefree values of p olynomial discriminants i. arXiv preprint arXiv:1611.09806, 2016

work page Pith review arXiv 2016
[2]

Bombieri

E. Bombieri. Le grand crible dans la th´ eorie analytique des nombre s. Ast´ erisque, (18):103, 1987

work page 1987
[3]

R. Brauer. A note on zeta-functions of algebraic number ﬁelds. Acta Arith., 24:325–327, 1973

work page 1973
[4]

Zeros of Rankin-Selberg $L$-functions at the edge of the critical strip

F. Brumley, J. Thorner, and A. Zaman. Zeros of Rankin-Selberg L-functions at the edge of the critical strip. arXiv e-prints , page arXiv:1804.06402, Apr 2018

work page Pith review arXiv 2018
[5]

W. Duke. Some problems in multidimensional analytic number theory . Acta Arith., 52(3):203–228, 1989

work page 1989
[6]

W. Duke. Bounds for arithmetic multiplicities. In Proceedings of the International Congress of Mathe- maticians, Vol. II (Berlin, 1998) , number Extra Vol. II, pages 163–172, 1998. A ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS 23

work page 1998
[7]

J. S. Ellenberg and A. Venkatesh. Reﬂection principles and bound s for class group torsion. Int. Math. Res. Not. IMRN , (1):Art. ID rnm002, 18, 2007

work page 2007
[8]

P. X. Gallagher. A large sieve density estimate near σ = 1. Invent. Math. , 11:329–339, 1970

work page 1970

Show all 23 references

[9]

D. R. Heath-Brown. Convexity bounds for L-functions. Acta Arith., 136(4):391–395, 2009

2009
[10]

Kowalski and P

E. Kowalski and P. Michel. Zeros of families of automorphic L-functions close to 1. Paciﬁc J. Math. , 207(2):411–431, 2002

2002
[11]

J. C. Lagarias and A. M. Odlyzko. Eﬀective versions of the Cheb otarev density theorem. In Algebraic number ﬁelds: L-functions and Galois properties (Proc. Sympos., Univ. Dur ham, Durham, 1975) , pages 409–464. Academic Press, London, 1977

1975
[12]

N. J. H. Lai and L. Silberman. A reﬁnement of the Aramata–Bra uer theorem. preprint, 2019

2019
[13]

Lau and J

Y.-K. Lau and J. Wu. A density theorem on automorphic L-functions and some applications. Trans. Amer. Math. Soc. , 358(1):441–472, 2006

2006
[14]

U. V. Linnik. On the least prime in an arithmetic progression. Rec. Math. [Mat. Sbornik] N.S. , 15(57):139–178,347–368, 1944

1944
[15]

M. R. Murty and V. K. Murty. A variant of the Bombieri-Vinograd ov theorem. In Number theory (Montreal, Que., 1985) , volume 7 of CMS Conf. Proc. , pages 243–272. Amer. Math. Soc., Providence, RI, 1987

1985
[16]

M. R. Murty and V. K. Murty. Non-vanishing of L-functions and applications . Modern Birkh¨ auser Classics. Birkh¨ auser/Springer Basel AG, Basel, 1997. [2011 repr int of the 1997 original] [MR1482805]

1997
[17]

M. R. Murty, V. K. Murty, and N. Saradha. Modular forms and t he Chebotarev density theorem. Amer. J. Math. , 110(2):253–281, 1988

1988
[18]

Neukirch

J. Neukirch. Algebraic number theory , volume 322 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]
[19]

L. B. Pierce, C. L. Turnage-Butterbaugh, and M. Matchett W ood. On a conjecture for ℓ-torsion in class groups of number ﬁelds: from the perspective of moments. arXiv e-prints , Feb 2019

2019
[20]

L. B. Pierce, C. L. Turnage-Butterbaugh, and M. Matchett W ood. An eﬀective Chebotarev density theorem for families of number ﬁelds, with an application to ℓ-torsion in class groups. Invent. Math. , accepted for publication
[21]

Soundararajan and J

K. Soundararajan and J. Thorner. Weak subconvexity withou t a Ramanujan hypothesis. Duke Math. J., 168:1231–1268, 2019. With an appendix by Farrell Brumley

2019
[22]

H. M. Stark. Some eﬀective cases of the Brauer-Siegel theor em. Invent. Math. , 23:135–152, 1974

1974
[23]

Thorner and A

J. Thorner and A. Zaman. A uniﬁed and improved Chebotarev de nsity theorem. Algebra Number Theory, 13(5):1039–1068, 2019. Department of Mathematics, University of Florida, Gainesv ille, FL 32611 E-mail address : jthorner@ufl.edu Department of Mathematics, University of Toronto...

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Squarefree values of polynomial discriminants I

M. Bhargava, A. Shankar, and X. Wang. Squarefree values of p olynomial discriminants i. arXiv preprint arXiv:1611.09806, 2016

work page Pith review arXiv 2016

[2] [2]

Bombieri

E. Bombieri. Le grand crible dans la th´ eorie analytique des nombre s. Ast´ erisque, (18):103, 1987

work page 1987

[3] [3]

R. Brauer. A note on zeta-functions of algebraic number ﬁelds. Acta Arith., 24:325–327, 1973

work page 1973

[4] [4]

Zeros of Rankin-Selberg $L$-functions at the edge of the critical strip

F. Brumley, J. Thorner, and A. Zaman. Zeros of Rankin-Selberg L-functions at the edge of the critical strip. arXiv e-prints , page arXiv:1804.06402, Apr 2018

work page Pith review arXiv 2018

[5] [5]

W. Duke. Some problems in multidimensional analytic number theory . Acta Arith., 52(3):203–228, 1989

work page 1989

[6] [6]

W. Duke. Bounds for arithmetic multiplicities. In Proceedings of the International Congress of Mathe- maticians, Vol. II (Berlin, 1998) , number Extra Vol. II, pages 163–172, 1998. A ZERO DENSITY ESTIMATE FOR DEDEKIND ZETA FUNCTIONS 23

work page 1998

[7] [7]

J. S. Ellenberg and A. Venkatesh. Reﬂection principles and bound s for class group torsion. Int. Math. Res. Not. IMRN , (1):Art. ID rnm002, 18, 2007

work page 2007

[8] [8]

P. X. Gallagher. A large sieve density estimate near σ = 1. Invent. Math. , 11:329–339, 1970

work page 1970