A secondary term in the first moment of primitive cubic Dirichlet L-functions

Ziwei Hong

REVIEW

A secondary term in the first moment of primitive cubic Dirichlet L-functions

Ziwei Hong

Number Theory · Submitted 1 Nov 2026

Pith's verdict: First explicit secondary term for the non-Kummer cubic first moment, coherent and worth refereeing, but the printed normalization of the imported residue in Lemma 2.2 is ambiguous enough that the coefficient needs checking.

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arXiv ID2611.01451 (version 2)

Pith IDH36FWPXR, Pith's permanent record of this paper and its review

CitationZiwei Hong. "A secondary term in the first moment of primitive cubic Dirichlet L-functions." arXiv:2611.01451 (2026). https://pith.science/paper/H36FWPXR

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Editor's note

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Hong proves, under GRH, that the smoothed first moment of primitive cubic Dirichlet L-functions over Q has main term C3 Phi(1)Q + C'_3 Phi(5/6)Q^{5/6} + O(Q^{23/28+eps}), with C'_3 explicit and negative. That is new: the earlier secondary-term results were for Kummer cubic or quadratic Hecke families, and Gao-Zhao stopped at Q^{7/8}. The method is clean: use Gao-Zhao's double Dirichlet series, import bounds and residues for the Gauss-sum series from David-de Faveri-Dunn-Stucky, continue via Bochner's tube theorem and the comparison principle, and read off the pole at s=5/6. There are no fitted parameters, no invented entities, and no circularity. The Mellin identity (3.18) is exact, the exponent arithmetic checks (5/6 - 23/28 = 1/84), and the sign of C'_3 follows because zeta(5/6)<0 and the remaining factors are positive. This deserves a serious referee. The soft spots are real but localized. The theorem rests on Lemma 2.1 and Lemma 2.2, both imported from the unpublished preprint [3]. Lemma 2.2 is the residue at s=4/3, and C'_3 is exactly that residue. The proof as printed is internally ambiguous: it states the unrestricted residue from [3] is c0 tau3(r) N(r)^{1/6} and then says, after division by N(r)^{1/6}, the relative ramified factors are 1, 3^{-1/3}, 0. Read literally, multiplying by N^{1/6} with tau3(1)=tau3(3)=27 gives relative factors 1 and 3^{1/3}, not 1 and 3^{-1/3}; formula (2.7) asserts 3^{-1/3}. The two are consistent only if the intended [3] residue is c0 tau3(r)/N(r)^{1/6}, or if the division is applied before tau3 is absorbed. Since the ramified local factor at 3 in (3.25) is built from this residue, an exponent slip changes the numerical value of C'_3 by a factor of 3^{2/3}. The sign survives, and I do not see a structural fault in the rest of the argument, but the advertised coefficient is exactly what cannot be checked from this paper alone. The author does not hide the reliance on [3]; it is explicit. But for a theorem whose headline is an explicit constant, that dependency is load-bearing. The right audience is people working on moments of L-functions and multiple Dirichlet series; they will find a useful way to get secondary terms without constructing and sieving corrected multiple Dirichlet series. A serious referee should check Lemma 2.2 against [3] before anything else. I would send this to review rather than desk-reject it.

Referee report

Significance

If the coefficient C'_3 is correct, the result is a meaningful, explicit advance in the first moment problem for cubic Dirichlet L-functions: it isolates a lower-order term that the GLH-based argument only sees as part of the error, and it does so with a comparatively simple double-Dirichlet-series argument that avoids the corrected multiple Dirichlet series construction in [4]. The internal structure is coherent in its main lines: the Mellin identity (3.18) is exact, the local Euler factors in (3.24)-(3.25) check out, and the exponent arithmetic 5/6 - 23/28 = 1/84 is correct. The proof is not circular: the secondary coefficient is obtained from a residue computation rather than assumed or fitted. However, the explicit value of C'_3 depends on residue formulas imported from an unpublished preprint, and the proof of Lemma 2.2 as printed contains an internal inconsistency; until that is resolved, the numerical coefficient in (1.4) is not established.

Major objections (2)

The conversion from the trace normalization of [3] to the c_K normalization is not internally consistent as printed. The proof states that the unrestricted residue in [3, Lemma 6.3] is c0 tau3(r) N(r)^{1/6} and then says that after division by N(r)^{1/6} the relative ramified factors are 1, 3^{-1/3}, 0. With tau3(1)=tau3(3)=27, the first formula gives residues proportional to 1 and 3^{1/3} for r=1 and r=3, respectively, whereas (2.7) asserts factors 1 and 3^{-1/3}. The two statements agree only if the intended formula is c0 tau3(r) N(r)^{-1/6}, or if the division by N(r)^{1/6} is performed before the residue constant is read off. This is load-bearing: the nu=1 residue is the input for the ramified local factor 1+3^{-w-1/3}/(1-3^{-3w}) in (3.25) and therefore determines the explicit value of C'_3 in (1.4). The proof should display the residue formula from [3] correctly and spell out the normalization change leading to (2.7).
The key analytic inputs (2.4) and (2.7) are imported from [3], which is cited as an arXiv preprint (arXiv:2410.03048v3, 2026) and is not a peer-reviewed publication. Lemma 2.1 and Lemma 2.2 are not re-derived in this paper, yet Theorem 1.1 is exactly as strong as the validity of these imported formulas after the normalization change (2.1). Because the central secondary coefficient is obtained from the residue (2.7), the dependence on an unpublished source is a substantive correctness risk rather than a routine citation. The authors should either include a self-contained verification of Lemma 2.2, in particular the conversion of the residue under the normalization change, or update the reference once [3] is published; at present the reader cannot check this step from the manuscript alone.

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The paper

The paper claims

Under GRH, the smoothed first moment of primitive cubic Dirichlet L-functions is proved to equal a main term of size $Q$ plus an explicit, provably negative secondary term of size $Q^{5/6}$, with error $O(Q^{23/28+\varepsilon})$.

The reading

This paper proves that the smoothed first moment of primitive cubic Dirichlet L-functions has a genuinely two-term asymptotic. Averaging $L(1/2,\chi)$ over primitive characters of exact order three with conductor $q\sim Q$, weighted by a smooth bump $\Phi$, gives under the generalized Riemann hypothesis $C_3\hat\Phi(1)Q + C'_3\hat\Phi(5/6)Q^{5/6} + O(Q^{23/28+\varepsilon})$. The earlier treatment [5] had the $Q$ main term with error $O(Q^{7/8+\varepsilon})$; the new content is the explicit $Q^{5/6}$ term together with a sharper error. Because $5/6 - 23/28 = 1/84$, choosing $\varepsilon < 1/84$ separates the secondary term from the error, and the coefficient $C'_3$ is displayed as a convergent Euler product that is provably negative. A reader should care because explicit lower-order terms in moment asymptotics are rarely isolated, and here the entire secondary term is traced to a single pole of a double Dirichlet series, making the arithmetic mechanism fully visible.

What carries the argument

The central object is the double Dirichlet series $A_3(s,w)$ of [5] — the generating series of values $L(w,\chi)$ over order-three characters weighted by $q^{-s}$ — together with its functional-equation partner $C_{3,1}(s,w)$, a ratio of gamma factors times a series of cubic Gauss sums. Cubic reciprocity and the evaluation $g_K(c_K,\chi_{3,d})=d$ reduce $C_{3,1}$ to a weighted sum over integers $m,d$ of the Dirichlet series of cubic Gauss sums $H_{3,1}(s, m d\, c_K)$, defined over $K=\mathbb{Q}(\sqrt{-3})$; this is the object whose poles carry the secondary term. Two inputs do the analytic work: Lemmas 2.1 and 2.2, which import from [3] a growth bound and the exact residue formula (2.7) for $H_{3,1}(s, r c_K)$ after the additive normalization change in (2.1), and Lemma 3.1, which estimates the $m,d$ sums under GRH through a ratio of L-functions. A three-lines interpolation transfers endpoint bounds across the strip, and Bochner's tube theorem extends the continuation to $\Re s > 23/28$ at $w=\tfrac12$. The secondary coefficient then emerges as the sum over $m,d$ of the residue (2.7), evaluated in closed form by (3.25), and Remark 3.3 verifies the normalization by comparing residues at the intersection point $(s,w)=(1,\tfrac13)$.

Extended reading notes

Core claim

The claim, stated as Theorem 1.1, is that under GRH for Dirichlet L-functions and for Hecke L-functions over $K = \mathbb{Q}(\sqrt{-3})$, the smoothed first moment over primitive cubic characters satisfies $$\sum_{(q,3)=1}\;\sum_{\substack{\chi\bmod q\\ \$chi^{3}$=\chi_0}}^{*}\!\! L(\tfrac12,\chi)\,\Phi(q/Q) = C_3\hat\Phi(1)Q + C'_3\hat\Phi(\tfrac56)$Q^{{5/6}}$ + O_{\Phi,\varepsilon}($Q^{{23/28+\varepsilon}}$).$$ The leading constant $C_3$ is the one obtained by [5] at the central point; the new constant $C'_3$, displayed in (1.4), is a product of $(2\pi)^{5/3}$, $\zeta(3/2)$, $\zeta(5/6)$, a gamma factor divided by $\zeta_K(2)$, the ramified factor $1-3^{-5/3}$, and two absolutely convergent Euler products over primes $p\equiv 1$ and $p\equiv 2 \pmod 3$. Every factor is positive except $\zeta(5/6)<0$, so $C'_3<0$. The proof's content is analytic continuation: the double Dirichlet series $A_3(s,\tfrac12)$ is shown to be meromorphic for $\Re s > 23/28$ with at most simple poles at $s=1$ and $s=5/6$ and polynomial growth between them, so the Mellin integral defining the moment can be shifted to $\Re s = 23/28+\varepsilon$, and the residue at $s=5/6$ — the trace of the polar line $s+w=4/3$ through the Gauss-sum series $H_{3,1}$ — is computed in closed form and equals $C'_3$.

Load-bearing premise

The whole $Q^{5/6}$ term is read off formula (2.7), the residue of the Gauss-sum series $H_{3,1}$ at $s=4/3$ as imported from [3] under the normalization change in (2.1): if that residue is wrong under this paper's convention — a step the paper flags as necessary but does not re-derive — the secondary coefficient changes size or sign, whereas the assumed Riemann hypotheses would only affect the size of the error term.

Editorial extensions

If this is right

Because $C'_3<0$ while $\hat\Phi(5/6)>0$ for any nonzero weight, the average central value provably falls below its leading approximation $C_3\hat\Phi(1)Q$, by an amount of order $Q^{5/6}$ that dominates the error whenever $0<\varepsilon<1/84$.
The secondary coefficient is an absolutely convergent product of local factors, so the $Q^{5/6}$ term is a fully explicit local-global statement: primes $p\equiv 1$ and $p\equiv 2 \pmod 3$ contribute separate Euler factors, and only the prime 3 enters through the global zeta value $\zeta(5/6)$.
The residue formula (2.7) gives a precise arithmetic description of which twists contribute to the secondary term: a square part $b^2$ in the twist forces a zero theta coefficient unless $b=1$, so only squarefree $a$ and cube factors survive, with a factor $\kappa_p$ for each covered prime.
Taken with [5], the result fixes three mutually resolvable scales for this moment — $Q$, $Q^{5/6}$, and $O(Q^{23/28+\varepsilon})$ — so the asymptotic is not just structural but fully determined numerically at each scale.

Reading between the lines

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

The same polar-divisor mechanism predicts explicit secondary terms in other fixed-order families over $\mathbb{Q}$ whenever a double Dirichlet series has a divisor crossing the central line $w=\tfrac12$; the paper stops at cubic characters, and quartic or sextic analogues would need only the analogous Gauss-sum residue.
The negativity of $C'_3$ is imposed by the global factor $\zeta(5/6)<0$ while every local Euler factor is positive, suggesting that the sign of such secondary coefficients is a global, not a local, phenomenon — a pattern worth testing in other families where a secondary coefficient is known or computable.
A direct numerical probe of the theorem's sharpest content is feasible: after subtracting the main term from a computed smoothed first moment, the residual should track $C'_3\hat\Phi(5/6)Q^{5/6}$ with the predicted negative sign, so even a moderate-precision estimate of the residual would test the sign, which all local factors alone cannot determine.

Original abstract

P. Gao and L. Zhao obtained an asymptotic formula for the smoothed first moment of primitive cubic Dirichlet $L$-functions over $\Q$ with conductors coprime to $3$, with a main term of order $Q$ and error $O\left(Q^{\frac{7}{8}+\eps}\right)$ under the generalized Lindel\"of hypothesis. Building on their double Dirichlet series, we assume the generalized Riemann hypothesis and separate an explicit secondary term of order $Q^{\frac{5}{6}}$ in the same family, with error $O\left(Q^{\frac{23}{28}+\eps}\right)$. We explicitly evaluate the secondary coefficient and prove that it is strictly negative. We also prove unconditionally that the secondary polynomial in the squarefree second moment of the natural cubic Hecke $L$-functions over $\Q(\sqrt{-3})$ has degree one and strictly positive leading coefficient, which we evaluate explicitly.

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

P. Gao and L. Zhao obtained an asymptotic formula for the smoothed first moment of primitive cubic Dirichlet $L$-functions over $Q$ with conductors coprime to $3$, with a main term of order $Q$ and error $O\left(Q^{\frac{7}{8}+\eps}\right)$ under the generalized Lindel\"of hypothesis. Building on their double Dirichlet series, we assume the generalized Riemann hypothesis and separate an explicit secondary term of order $Q^{\frac{5}{6}}$, with error $O\left(Q^{\frac{23}{28}+\eps}\right)$.

background contribution method result interpretation main claim

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s0	P. Gao and L.	background 0.98	0.03	0.01	unclear 1.00
s1	Zhao obtained an asymptotic formula for the smoothed first moment of primitive cubic Dirichlet $L$-functions over $Q$ with conductors coprime to $3$, with a main term of order $Q$ and error $O\left(Q^{\frac{7}{8}+\eps}\right)$ under the generalized Lindel\"of hypothesis.	background 1.00	0.04	0.07	asserted 1.00
s2	Building on their double Dirichlet series, we assume the generalized Riemann hypothesis and separate an explicit secondary term of order $Q^{\frac{5}{6}}$, with error $O\left(Q^{\frac{23}{28}+\eps}\right)$.	result 0.55	0.94	0.85	asserted 0.97

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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