Simultaneous nonvanishing of Dirichlet $L$-functions in Galois orbits

Alexandra Florea; Hieu T. Ngo; Hung M. Bui

REVIEW 3 major objections 3 minor 43 references

Simultaneous nonvanishing of Dirichlet $L$-functions in Galois orbits

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Under GRH, a positive proportion of characters in every Galois orbit satisfy simultaneous nonvanishing of two twisted Dirichlet L-functions.

desk verdict The simultaneous nonvanishing theorem rests on an unconditional twisted-moment estimate whose off-diagonal reduction uses the wrong congruence and whose main term drops a nonzero orbit-average factor. read the letter →

arxiv 2507.06609 v1 pith:2QGRSE2B submitted 2025-07-09 math.NT

Hung M. Bui , Alexandra Florea , Hieu T. Ngo This is my paper · ORCID

classification math.NT MSC 11M0611M2611N37

keywords DirichletL-functionssimultaneousnonvanishingGaloisorbitsEulerproductmollifiermollifiedmomentsGeneralizedRiemannHypothesisp-adicRoththeorem

topics The Riemann Hypothesis

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 that, assuming the Generalized Riemann Hypothesis, for any fixed odd prime p and any Galois orbit of primitive Dirichlet characters modulo q=p^k, a positive proportion of characters in the orbit satisfy L(1/2,χη1)L(1/2,χη2)≠0 for any two distinct imprimitive characters η1,η2. This is the first simultaneous nonvanishing result inside Galois orbits, extending an earlier single-function nonvanishing proportion. The proof works by bounding a mollified fourth moment from above via a new Euler product mollifier and a mollified second moment from below, with the lower bound relying on p-adic Diophantine approximation. If correct, the result shows the two central values are jointly nonvanishing for a positive share of the orbit as k→∞.

What carries the argument

The Euler product mollifier M(χ)=∏_{j=0}^K E_{ℓ_j}(-P_{I_j}(χ;K)) built from prime sums over dyadic intervals (equations (30)–(31)), which molifies the central values by truncating a Dirichlet polynomial with a completely multiplicative weight a(n;K). The argument pairs it with the orbit character averages E_O[χ(n)] from the prior framework, which vanish unless $n^{{p-1}}$≡1 (mod $p^{{k-1}}$), converting off-diagonal terms into p-adic congruence sums D_0,D_1,D_2. These sums are bounded by elementary estimates plus the p-adic Roth theorem, giving the power-saving error in the twisted second moment. The fourth moment bound (Theorem 1.3) then follows from the conditional log-moment inequality of Chandee with the sharp 2v-th moment bound, and the second moment lower bound uses Granville–Soundararajan's lemma for log L(1,η) under GRH.

What would settle it

A concrete check: verify numerically, for a fixed small odd prime p and increasing k, whether the twisted second moment average (1/|O|) Σ_{χ∈O} L(1/2,χη1)L(1/2,χη2) equals L(1,η1η2) to the error claimed in Corollary 1.6; a persistent discrepancy beyond the stated power saving would invalidate Theorem 1.5 and therefore Theorem 1.1. Alternatively, an explicit counterexample with a Galois orbit and two distinct imprimitive characters for which the nonvanishing count is o(q) would refute the main theorem.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: under GRH, the count of χ in a fixed Galois orbit O for which L(1/2,χη1)L(1/2,χη2)≠0 is ≫ q, i.e., a positive proportion of the orbit size. The key structural discovery is that small-character twisting preserves primitivity (χη is primitive for primitive χ and imprimitive η), and that the twisted second moment over an orbit has a main term pinned down by L(1,η1η2) with power-saving error, unconditionally (Theorem 1.5). The GRH enters to dominate the fourth moment of L(1/2,χ)M(χ) by O(q) and to force the factor |L(1,η1η2)| in the mollified second moment to be bounded below.

Load-bearing premise

The argument assumes the Generalized Riemann Hypothesis for Dirichlet L-functions; without it the upper bound on the mollified fourth moment and the lower bound on the mollified second moment both break down.

Editorial extensions

If this is right

For any fixed odd prime p, all Galois orbits modulo p^k get simultaneous nonvanishing with a uniform positive proportion as k→∞.
The unconditional second moment along a full orbit equals L(1,η1η2) plus a power-saving error, giving a precise average of the product of the two central values.
An unconditional weaker simultaneous nonvanishing bound of ≫ε q^{2/3−ε} follows via the Weyl subconvexity bound, so the phenomenon does not depend entirely on GRH.
Along thinner orbits (subquotients of the cyclotomic tower) the same twisted moment asymptotic holds under compatibility conditions, giving unconditional second moments there.
The sharp mollified fourth moment bound actually holds for all q→∞, not only prime powers, so the mollifying technique may apply to broader families.

Reading between the lines

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

The Euler product mollifier construction could be transferred to the family of cubic Hecke L-functions over Galois orbits, where the current nonvanishing densities are weaker; the obstruction is the analogue of the L(1,η) lower bound.
The p-adic congruence counting (Proposition 2.15) is likely to be reusable wherever orbit averages force congruences mod p^α, e.g., in higher twists or ray class characters over p-adic towers.
Replacing GRH by a sufficiently strong zero-free region below 3/4 might keep Theorem 1.5 with a smaller saving, suggesting the critical loss is only the explicit log L(1,η) expansion.
The compatibility window for thin orbits indicates a quantitative threshold in κ that could be probed numerically, possibly predicting where the second moment ceases to be dominated by the diagonal term.

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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

The main theorem is not established as written. The problem sits in Section 3.1, in the proof of Theorem 1.5, the unconditional twisted second moment that everything else rests on. A stress-test objection I checked holds up: the off-diagonal sums are reduced to the wrong congruence. Lemma 2.3 says E_O[chi(n)] is nonzero only if n^{p-1} = 1 (mod p^{k-1}). Applied to n = m1 n1 m2 n2, the support is (m1 n1 m2 n2)^{p-1} = 1, i.e. ab = zeta (mod p^{k-1}) with a = m1 n1 and b = m2 n2. The paper instead restricts to a^{p-1} = b^{p-1}, i.e. a = zeta b. These differ: for p = 3, modulus 27, a = 2, b = 14 gives ab = 1, so (ab)^2 = 1, but a^2 = 4 and b^2 = 7. The D0(A,B;p^{k-1}) sums in (16) therefore neither contain nor bound the actual off-diagonal contribution. The same error affects E- with modulus p^{k-h}, and the thin-orbit Theorem 1.9 inherits it. This is load-bearing: Theorem 1.5 feeds Theorem 1.2, which feeds Theorem 1.1.

A second, related problem: the diagonal main term (14) replaces E_O[chi(m1 n1 m2 n2)] by 1 when m1 n1 = m2 n2, but E_O[chi(t^2)] is not 1 - for the order-18 orbit modulo 27, E_O[chi(4)] = 0. So the asymptotic (15) is not the actual diagonal contribution as written. The authors would need to show that the orbit-average support condition does not change the residue.

Credit where it is due: the question is natural and new - simultaneous nonvanishing in Galois orbits, where prior work only had non-simultaneous nonvanishing. The Euler product mollifier machinery in Section 4 (Theorem 1.3, Lemmas 4.3-4.4) is developed in real detail and reads coherently; the Holder step in Section 6 is fine given Theorems 1.2 and 1.3. The thin-orbit second moments and Proposition 2.15 are genuine additions. The reliance on GRH and Roth-Ridout is explicit, and the heavy use of [18] is legitimate even though the adaptation of that work is exactly where the error crept in.

The reader's CONDITIONAL verdict rested on the sketched Proposition 4.2 and the conditional input to Theorem 1.3 - real but secondary. The deeper problem is in Section 3.1.

Recommendation: send it to a serious referee anyway. The problem and the machinery deserve scrutiny, and the flaw may be repairable - the correct support ab = zeta is the natural place to look. But I would not cite the main result as established.

Referee Report

3 major / 3 minor

Summary. The paper claims that, under GRH, for a fixed odd prime p and q=p^k, given any two distinct imprimitive Dirichlet characters eta1, eta2 modulo q, a positive proportion of primitive characters chi in any fixed Galois orbit O (as k grows) satisfy L(1/2, chi eta1) L(1/2, chi eta2) != 0. The proof is based on a mollified second moment lower bound (Theorem 1.2) and a mollified fourth moment upper bound (Theorem 1.3), with the unconditional twisted moment asymptotic (Theorem 1.5) as the basis for the lower bound. The paper also proves unconditional second moment asymptotics for full and thin Galois orbits (Corollaries 1.6 and 1.11).

Significance. If the main theorem were correct, it would be a meaningful extension of simultaneous nonvanishing results to the refined family of Galois orbits of Dirichlet characters, building on the work of Khan, Milicevic, and Ngo. The use of an Euler-product mollifier and the Roth-Ridout theorem in this context is interesting, and the unconditional twisted-moment results would be of independent value. However, the proof of the central unconditional moment asymptotic (Theorem 1.5) contains what appear to be serious gaps in both the diagonal and off-diagonal contributions, and those gaps undermine the conditional main theorem as well.

major comments (3)

[§3.1, proof of Theorem 1.5, equations (12)–(16)] The support condition for the off-diagonal sums is misstated. Lemma 2.3 gives EO[chi(n)] != 0 only if n^{p-1} ≡ 1 mod p^{k-1}. In S+(m1,m2) one has n = m1 n1 m2 n2, so setting a = m1 n1 and b = m2 n2, the nonzero orbit-average condition is (ab)^{p-1} ≡ 1 mod p^{k-1}, i.e. a^{p-1} ≡ b^{-(p-1)} mod p^{k-1}. The paper instead reduces to sums with (m1 n1)^{p-1} ≡ (m2 n2)^{p-1} mod p^{k-1}, i.e. a^{p-1} ≡ b^{p-1} mod p^{k-1}. These two conditions are not equivalent; a concrete example is p=3, alpha=3, a=2, b=14, where ab ≡ 1 mod 27 but a^2 != b^2 mod 27. Thus the reduction to D0(A,B;p^{k-1}) as defined in (6) is not justified, and the bounds in (16)–(18) do not follow. The same issue affects E−, where the analogous condition is (m1 n2 m2 n1)^{p-1} ≡ 1 mod p^{k-h} rather than the stated same-root condition. Since this step is the core of the unconditional Theorem 1.5, it is load-bearing.
[§3.1, equation (14)] The main term M(m1,m2) is not the actual diagonal contribution. In S+(m1,m2) the summand contains the orbit-average factor EO[chi(m1 n1 m2 n2)]. For m1 n1 = m2 n2, this factor is EO[chi((m1 n1)^2)], not 1. For m1=m2=1, the diagonal sum is sum_r eta(r)/r V(pi r^2/(qX)) EO[chi(r^2)]. By Lemma 2.3, EO[chi(r^2)] != 0 only if r^{2(p-1)} ≡ 1 mod p^{k-1}; since -1 is not a (p-1)-th power modulo an odd prime power, this forces r^{p-1} ≡ 1 mod p^{k-1}, i.e. r lies in one of the p-1 Teichmuller residue classes modulo p^{k-1}. The resulting diagonal contribution is then O(log q / p^{k-1}) = o(1) as k tends to infinity, whereas the paper's M(1,1) equals L(1,eta) + o(1), which is typically of size 1. Thus the proposed main term in (14) is not merely missing a factor; it appears to be of the wrong order of magnitude. The paper does not explain why the omitted orbit-average factor can be replaced by 1.
[Theorems 1.2 and 1.1] Because Theorem 1.5 is the unconditional basis for the mollified second moment lower bound in Theorem 1.2, and Theorem 1.1 uses Theorem 1.2 together with the fourth moment bound, the gaps in the proof of Theorem 1.5 are load-bearing for the paper's main nonvanishing result. The proof as written does not establish the claimed asymptotics, and the issues with the main term suggest that the stated main term in Theorem 1.5 may be incorrect rather than merely under-proven. Further, the off-diagonal bounds would need to be replaced by bounds for the inverse-product congruence (ab)^{p-1} ≡ 1, for which the Roth-Ridout estimates in Proposition 2.15 do not directly apply as stated.

minor comments (3)

[Proposition 4.2] The proof of Proposition 4.2 is only sketched by reference to a similar argument in [9]. Since this proposition is an essential part of the mollified fourth moment bound, a fuller proof would be helpful, though this is secondary to the issues in Section 3.
[Notation] The notation X* is used both for sums over integers not divisible by p and for sums over primitive characters; this is noted in the text but is a potential source of confusion.
[Introductory remarks] There are several typos and minor errors in the introduction and remarks, e.g., 'Hecker' for 'Hecke' and 'neccesarily' for 'necessarily'; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation follows a standard moment method with external, independently published inputs.

full rationale

The main results are obtained by bounding a mollified second moment from below and a mollified fourth moment from above. Theorem 1.2 is a lower bound for a mollified second moment whose main term is L(1, eta1 eta2) times an explicit Euler product; Theorem 1.3 is an upper bound for a mollified fourth moment proved from a conditional moment bound of Soundararajan, Harper, and Szabo. Neither estimate is defined in terms of the nonvanishing count that Theorem 1.1 derives: the final Holder step combines the two estimates, and the nonvanishing count is not an input to either. The cited lemmas from [18] (Lemmas 2.3 through 2.11) are prior published results used for character averages and congruence estimates, not for the target simultaneous nonvanishing statement; although [18] shares an author, no uniqueness theorem or unverified premise is imported from that work to force the conclusion. Lemma 4.1, Proposition 4.2, and Lemmas 4.3 and 4.4 are the paper's own mollifier estimates, built on external tools such as Chandee, Granville-Soundararajan, and Szabo. No fitted constants are renamed as predictions; the parameters beta_j, s_j, and ell_j are chosen from inequalities with absolute constants. The review note about the off-diagonal support congruence (a^{p-1} congruent to b^{p-1} versus (ab)^{p-1} congruent to 1) is a possible correctness gap in the proof of Theorem 1.5, not a circular reduction of the theorem to its own assumptions, so it does not affect the circularity score.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The proof does not fit any constant to data. The free parameters listed are internal mollifier parameters chosen by hand; the main theorem is qualitative in q and does not depend on their precise values. GRH and Roth-Ridout are the main external inputs. No new particles, forces, dimensions, or other invented entities are introduced.

free parameters (3)

c (cutoff for beta_K) = 0 < c < 4(e^{1/4}-1)^4 / (e(v+2)^4)
Chosen by hand in equation (35) to make the hypotheses of Lemma 4.3 hold; the final positive proportion of nonvanishing does not depend on its exact value.
lambda = Solution of e^{-lambda} = lambda + lambda^2/2, approximately 0.317
A fixed constant from Chandee's explicit upper bound in Lemma 4.1, not fitted to any data.
ell_j and s_j = s_j = 2 floor(1/(8 beta_j)), ell_j = floor(s_j^{3/4})
Smoothing and truncation parameters in the Euler product mollifier, chosen in equation (36) to balance error terms; the qualitative theorem does not depend on their precise values.

assumptions (4)

domain assumption Generalized Riemann Hypothesis
Assumed in Theorems 1.1 and 1.2; used in Section 4.1 through Chandee and Soundararajan-Harper moment estimates, and in Section 5 through [11, Lemma 8.2] to control log L(1, eta).
standard math Roth-Ridout theorem (p-adic Roth theorem)
Used in Lemmas 2.9 and 2.11 and Proposition 2.15 through [30] and [18] to control p-adic congruence sums; deep but proven externally.
domain assumption Conditional moment estimates for Dirichlet L-functions
The bound sum |L(1/2, chi)|^{2v} << q (log q)^{v^2} is imported from [42, Theorem 1] under GRH and is essential for Theorem 1.3.
standard math Approximate functional equation and character orthogonality
Standard tools used in Lemma 2.1 and Sections 3 through 5; not in question.

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Pith. "Pith review of Simultaneous nonvanishing of Dirichlet $L$-functions in Galois orbits." pith.science (2026). https://pith.science/paper/2QGRSE2B

@misc{pith2026250706609,
  author       = {Pith},
  title        = {Pith review of: Simultaneous nonvanishing of Dirichlet $L$-functions in Galois orbits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2QGRSE2B}},
  note         = {Machine review of arXiv:2507.06609}
}

abstract

Under the Generalized Riemann Hypothesis, we prove that given any two distinct imprimitive Dirichlet characters $\eta_1, \eta_2$ modulo $q=p^k$, a positive proportion of characters $\chi$ modulo $q$ in a fixed Galois orbit of primitive characters satisfies the nonvanishing property that $L(1/2,\chi \eta_1) L(1/2,\chi \eta_2) \neq 0$, as $k \to \infty$ (with $p$ fixed). Previously, only a positive proportion of nonvanishing result was available in Galois orbits (as opposed to simultaneously nonvanishing), due to work of Khan, Mili\'cevi\'c and Ngo. The main ingredients are obtaining a sharp upper bound on the mollified fourth moment over the Galois orbit using an Euler product mollifier, and obtaining a lower bound for the mollified second moment, which relies on using results from Diophantine approximation (such as the $p$-adic Roth theorem). We also unconditionally compute the second moments for $L$--functions associated to primitive Dirichlet characters in full orbits and thinner orbits.

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Pith tools

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

[1] [18]

R. Khan, D. Mili´ cevi´ c, and H. T. Ngo, Non-vanishing of Dirichlet L-functions in Galois orbits , Int. Math. Res. Not. (2016), no. 22, 6955–6978

work page 2016

[2] [1]

Torsion of Rational Elliptic Curves over the $\mathbb{Z}_p$-Extensions of Quadratic Fields

O. Avci, Torsion of rational elliptic curves over the Zp-extensions of quadratic fields , preprint, arXiv:2505.04149

work page Pith review arXiv

[3] [2]

Baier and M

S. Baier and M. P. Young, Mean values with cubic characters , J. Number Theory 130 (2010), 879–903

work page 2010

[4] [3]

H. M. Bui, Non-vanishing of Dirichlet L-functions at the central point , Int. J. Number Theory 8 (2012), 1855–1881

work page 2012

[5] [4]

H. M. Bui, N. Evans, S. Lester, and K. Pratt, Weighted central limit theorems for central values of L-functions, J. Eur. Math. Soc. 27 (2025), 2477–2529

work page 2025

[6] [5]

Chandee, Explicit upper bounds for L-functions on the critical line , Proc

V. Chandee, Explicit upper bounds for L-functions on the critical line , Proc. Amer. Math. Soc. 137 (2009), 4049–4063

work page 2009

[7] [6]

Chinta, Analytic ranks of elliptic curves over cyclotomic fields , J

G. Chinta, Analytic ranks of elliptic curves over cyclotomic fields , J. Reine Angew. Math. 544 (2002), 13–24

work page 2002

[8] [7]

Chowla, The Riemann hypothesis and Hilbert’s tenth problem , London and Glasgow: Blackie & Son Ltd

S. Chowla, The Riemann hypothesis and Hilbert’s tenth problem , London and Glasgow: Blackie & Son Ltd. XV, 119 pages, 1965

work page 1965