Central non-vanishing of Dirichlet $L$-functions

Andrew Pearce-Crump; Benjamin Durkan

REVIEW

Central non-vanishing of Dirichlet $L$-functions

Benjamin Durkan, Andrew Pearce-Crump

Number Theory · Submitted 5 Sep 2026

Pith's verdict: Impressive new uniform 3/8 bound, but the proof relies on two unpublished preprints and a complex local estimate that could be fragile.

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arXiv ID2609.06167 (version 3)

Pith IDRTL3WV3O, Pith's permanent record of this paper and its review

CitationAndrew Pearce-Crump, Benjamin Durkan. "Central non-vanishing of Dirichlet $L$-functions." arXiv:2609.06167 (2026). https://pith.science/paper/RTL3WV3O

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Editor's note

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The main result—3/8 non-vanishing for all admissible moduli—is a genuine improvement over Qin–Wu's 7/19, and the 5/13 for squarefree moduli is clean. The framework is sensible: reduce everything to fourth moments of Kloosterman sums via Corollary 4.10, then estimate those moments. The squarefree and prime-square cases (Theorems 8 and 9) are well-handled and don't raise red flags.

The soft spots are real. First, the odd-parity extension of the moment evaluation (Lemma 4.3, Proposition 4.4) is simply transferred from Qin–Wu's preprint [25] with a one-sentence remark; no derivation is given. A referee would need to check whether the sign and gamma factors work out for odd characters. Second, the engine of the uniform 3/8 theorem is Proposition 9.1, a lengthy case-by-case analysis of Kloosterman correlations at prime powers. The proof runs over many pages and involves delicate Gauss sum manipulations, 2-adic expansions, and alignment dichotomies. A single sign error or omitted boundary case (e.g., when a linear coefficient vanishes mod p) would break the estimate. The stress-test is right to flag this.

That said, the paper is written carefully, the claims are explicit, and the authors appear aware of the complexity. They cite two unpublished preprints [23,25] that handle some of the heavy lifting; this means the paper's correctness is partly contingent on those works holding up.

Who is this for? Number theorists working on non-vanishing of L-functions and Kloosterman sums. It deserves a serious referee—this is not a desk reject. But the referee should be prepared to dig into Proposition 9.1 and ask for a self-contained justification of the odd-parity moment, or at least a clearer dependence on [25].

What would settle it

Compute, for a small admissible modulus $q$ with both parity classes nonempty (say $q\le10^4$), the parity-separated second moment $M_2^{(j)}(q)$ from the approximate functional equation with the mollifier at $\theta_1=\theta_2=0.1$, and check whether $M_2^{(j)}(q)-(c_1+c_2)^2-c_1^2/\theta_1-c_2^2/\theta_2$ behaves like $O(\log\log q/\log q)$ for each parity $j$; a visible failure for $j=1$ would falsify the odd-parity transfer of Lemma 4.3. Separately, evaluate the local correlation claimed in (9.18) at a small powerful modulus such as $r_1=27$ or $r_1=256$ over pairs $(\Delta,h)$ with $G_0\mid h$: a single pair violating the bound $|\sum_\epsilon C_\epsilon(h;k_1,k_2;r_1)|\ll r_1^{3/2+\varepsilon}(\Delta,h,r_1)^{1/2}$ would break Proposition 9.1 and hence the uniform $3/8$ theorem.

Referee report

Major objections (3)

[Section 9.1, Eq. (9.18) and surrounding text] The estimate (9.18) is the engine of the uniform 3/8 theorem. The derivation of the divisibility condition 'G ≥ G0/64' is not fully justified when multiple branched prime factors are present. The paper states that at every branched factor the loss is at most 2^3 and that the two-primary part loses at most 2^6, then concludes G ≥ G0/64. However, for several odd prime branches the product of those losses could exceed 64 unless the loss factors are not multiplicative in the divisor condition. The authors should give a precise accounting of the multiplicative constant, or adjust the divisor condition to be uniform, and verify that the estimate (9.18) holds with the claimed power-saving exponent.
[Lemma 4.3 / Proposition 4.4] The odd-parity extension of the moment evaluation is transferred from [25, p.5] with only a brief statement. The cancellation of the q2 = 1 term for the sine phase and the residue calculation for odd parity are not carried out. Since the parity-wise lower bound in Theorem 1 depends on these moment formulae, the authors should provide a more detailed verification, at least outlining the contour shift and the treatment of the q2 = 1 term.
[Proposition 9.1, definition of H in (9.6)] The definition 'H | r♯_1 / rad(r♯_1)' is cryptic. The factorization Q = r1 r2 and the role of s = H r2 should be explained more clearly. The construction of the divisor s and its relationship to the primitive part of the conductor is critical for the bilinear estimate; the exposition would benefit from a concrete example or a more conceptual justification.

Minor objections (6)

[Abstract, page 1] Typo: 'suﬀiciently' should be 'sufficiently'.
[Proposition 3.3] The notation Kl_FGKM and the reference to [10, Proposition 3.2] could be clarified by stating the normalisation explicitly: Kl2(t;p) = p^{-1/2} S(1,t;p).
[Lemma 4.5] The hypotheses M1M2 < q^{1-η1} and N1N2 ≤ q^{1+ε} should be stated in the text of the lemma for completeness; currently they appear only in the proof.
[Page 10] The reduction of the mixed phase from e(t)+(-1)^j e(-t) to cos(2π t) for j=0 and sin(2π t) for j=1 is clear, but the reference to the odd-parity calculation in [25] should give a concrete equation or page number.
[Lemma 9.7] The exponents (9.76)–(9.81) involve many parameters; a brief derivation (or a note that they follow direct substitution) would improve readability.
[References] Reference [25] is a preprint (arXiv:2504.11916). The paper should indicate whether it has been updated or accepted, and whether any part of the argument relies on unpublished details.

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The paper

The paper claims

For every sufficiently large admissible modulus $q$, at least $3/8-\varepsilon$ of the primitive Dirichlet characters modulo $q$ have $L(1/2,\chi)\ne0$, with stronger proportions $5/13$ and $2/5$ under conditions on the powerful part.

The reading

At the centre of the critical strip, the Dirichlet $L$-function of every primitive character is conjectured to be nonzero, yet quantitative theorems only control a positive proportion of characters. This paper proves that for every sufficiently large modulus $q\not\equiv2\bmod4$ — the class that actually admits primitive characters — the proportion $\kappa(q)$ of primitive characters with $L(1/2,\chi)\ne0$ is at least $3/8-\varepsilon$, with no restriction on the prime-power factorization of $q$. The same proportion holds inside each parity class $\chi(-1)=\pm1$. The constant $3/8$ had previously been reached only for prime moduli, while the best bound valid for general conductors was $7/19-o(1)$, so the paper matches the prime-modulus record uniformly. It also proves $5/13-o(1)$ for squarefree moduli and for moduli whose powerful part satisfies $P(q)\le q^{1/2}$, and $2/5-o(1)$ when the powerful part has a divisor of size $q^{1/3+o(1)}$.

What carries the argument

The carrying object is the correlation of four Kloosterman sums over a complete residue system, $$A_q(B)=\sum_{\substack{1\le b_1,b_2,b_3,b_4\le B\\(b_1b_2b_3b_4,q)=1}}\left|\sum_{h\bmod q}S(h,b_1;q)S(h,b_2;q)S(h,b_3;q)S(h,b_4;q)\right|.$$ A two-piece mollifier with a root-number term turns the non-vanishing problem into bounds for $A_q(B)$: once the mixed cross terms are estimated, Cauchy's inequality gives $\kappa(q)\ge(\theta_1+\theta_2)/(1+\theta_1+\theta_2)+o(1)$, so the task is to maximise $\theta_1+\theta_2$ subject to the available estimates. Squarefree moduli are handled by multiplicativity and a count of exceptional pairings among the four arguments. Prime squares use an explicit evaluation of $S(a,b;p^2)$ as a sum over the two Hensel square roots, the exceptional configurations being exactly the sign relations among the four roots captured by a degree-12 polynomial congruence whose integer zeros are few and whose reductions are averaged over primes. For general moduli, a reciprocal bilinear estimate (Proposition 9.1) with local complete-correlation estimate (9.18) controls moduli with intermediate prime factors, and a Chinese-remainder operator bound (Proposition 9.5) built from the fourth Schatten norm plus a count of differences of square-root preimages (Lemma 9.3) controls moduli with a large prime square. The constraints become tight simultaneously at $w=1/3$, giving $2/5$, while the universal $3/8$ comes from the optimal split $q^{1/5}$ of the divisor construction.

Extended reading notes

Core claim

The central claim is Theorem 1: for every $\varepsilon>0$ and every sufficiently large integer $q\not\equiv2\bmod4$, at least $3/8-\varepsilon$ of the primitive Dirichlet characters modulo $q$ satisfy $L(1/2,\chi)\ne0$, and the same bound holds within each nonempty parity class $\chi(-1)=(-1)^j$. The point is uniformity: previously $3/8-o(1)$ had been attained only for prime conductors, and the best bound valid for all conductors was $7/19-o(1)$. The refinements are graded by the powerful part $P(q)=\prod_{\ell^\nu\|q,\ \nu\ge2}\ell^\nu$ of the modulus. Squarefree $q$, and more generally $P(q)\le q^{1/2}$, give $5/13-\varepsilon$; prime squares give $5/13-\varepsilon$ outside a sparse exceptional set of primes, surviving the adjunction of a fixed coprime factor; and moduli whose powerful part contains a divisor of size $q^{1/3+o(1)}$, in particular $q=rp^2$ with $p^{4-\eta}\le r\le p^{4+\eta}$, give the largest constant $2/5-\varepsilon$. Density-wise, all but $O(X^{3/4})$ of the moduli up to $X$ satisfy the stronger bound.

Load-bearing premise

The parity-wise form of Theorem 1 rests on a transfer taken from earlier work rather than derived here: swapping the even-parity weight $V_0$ for its odd-parity analogue $V_1$ in the moment evaluation is assumed to leave the main terms $1+1/\theta$ and $2c_1c_2$ unchanged (Lemma 4.3 and Proposition 4.4). A second load-bearing pillar is the local complete-correlation estimate (9.18) at the heart of Proposition 9.1, which drives the uniform $3/8$ bound for moduli whose powerful part has intermediate prime factors.

Editorial extensions

If this is right

For every sufficiently large admissible modulus $q$, at least $3/8-\varepsilon$ of the primitive characters modulo $q$ have non-vanishing central value, in each parity class separately, with no hypothesis on the factorization of $q$ (Theorem 1).
The stronger constant $5/13-\varepsilon$ holds for squarefree moduli and for all moduli whose powerful part is at most $q^{1/2}$, with only $O(X^{3/4})$ exceptions up to $X$ (Theorems 2 and 3).
Prime-square conductors reach $5/13-\varepsilon$ for all primes outside a set of size $O(X^{1-\xi_1}/\log X)$, a conclusion that extends to fixed multiples $mp^2$ and to varying short squarefree cofactors $r\le p^\alpha$ (Theorems 5–7).
Moduli whose powerful part has a divisor of size $q^{1/3+o(1)}$, including $q=rp^2$ with $p^{4-\eta}\le r\le p^{4+\eta}$, achieve the paper's largest constant $2/5-\varepsilon$ (Theorem 4 and Corollary 9.9).
Together the results give a dichotomy in terms of $\alpha=\log_q P(q)$: proportion $5/13-\varepsilon$ when $\alpha\le1/2$, and $3/8-\varepsilon$ in every case (Corollary 11.1).

Reading between the lines

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

If the paper's picture is right, the constant $3/8$ is a structural output of this mollifier family rather than a barrier imposed by one estimate: the paper shows the divisor construction's split point $q^{1/5}$ is optimal within the method and that the three constraint lines meet at $w=1/3$, so raising the universal constant would require a genuinely new input, not a retuning of the present one.
The graded results suggest the attainable proportion varies with where the divisors of $P(q)$ sit, interpolating between $5/13$ in the small-powerful-part regime and $2/5$ at the $q^{1/3}$ regime; intermediate moduli with powerful parts of medium size may admit a continuous spectrum of constants along the lines of the paper's Theorem 11 for mixed conductors.
The inequality $\kappa(q)\ge(\theta_1+\theta_2)/(1+\theta_1+\theta_2)+o(1)$ is the sharpest-looking bottleneck in the method, so the fourth-moment problem for the correlation $A_q(B)$ on powerful moduli is a natural next target: any improvement there would feed directly into a better universal proportion.
The odd-parity moment transfer is exactly the kind of step that decides whether parity-wise theorems hold at the same constants as their even counterparts; a self-contained derivation of the odd-parity mixed diagonal would make the parity-wise claims of Theorem 1 independent of the cited evaluation and would likely carry over to other conductors treated by the same mollifier.

Original abstract

We prove that, for every sufficiently large modulus $q\not\equiv2\bmod4$, at least $3/8-o(1)$ of the primitive Dirichlet characters $\chi$ modulo $q$ have $L(1/2,\chi)\ne0$. We also establish stronger proportions for moduli $q$ satisfying certain arithmetic conditions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,307 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.69 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.09 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We prove that, for every sufficiently large modulus $q\not\equiv2\bmod4$, at least $3/8-o(1)$ of the primitive Dirichlet characters $\chi$ modulo $q$ have $L(1/2,\chi)\ne0$. We also establish stronger proportions for moduli $q$ satisfying certain arithmetic conditions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We prove that, for every sufficiently large modulus $q\not\equiv2\bmod4$, at least $3/8-o(1)$ of the primitive Dirichlet characters $\chi$ modulo $q$ have $L(1/2,\chi)\ne0$.	result 0.97	0.97	0.93	asserted 1.00
s1	We also establish stronger proportions for moduli $q$ satisfying certain arithmetic conditions.	result 0.83	0.94	0.06	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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