REVIEW
Improved first moments of primitive quartic and sextic Dirichlet $L$-functions
Number Theory · Submitted 1 Nov 2026 · 8 references
Pith's verdict: New GRH-conditional error exponents with coherent interpolation geometry, but the central Euler-product identity is printed with L rather than 1/L and as written the proof does not go through.
T0 · AI review
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Referee report
The paper
The paper claims
Assuming the generalized Riemann hypothesis, first moments of primitive quartic and sextic Dirichlet $L$-functions have errors $O(Q^{49/58+\varepsilon})$ and $O(Q^{23/26+\varepsilon})$, improving $O(Q^{9/10+\varepsilon})$ and…
The reading
What carries the argument
The central object is the double Dirichlet series $A_j(s,w) = \sum_{q>1}\sum_\chi^* L(w,\chi)q^{-s}$ and its Gauss-sum reflected series $C_{j,\delta}(u,v)$, together with the generating series $H_\beta(u,r)$ of quartic and sextic Gauss sums. The key identity is Lemma 3.2 (quartic) and its analogue (4.15) (sextic): with the character extended by zero at the primes of its modulus, the Möbius-weighted sum $\sum_{m,d}\mu(d)\chi(m)\chi(d)^{j-2}m^{-z}d^{-t}$ factors exactly as $L(z,\chi)L^{(2)}(t,\chi^{j-2})$ times a convergent correction product, converting the removal of rational prime divisors into a ratio of $L$-functions. The second input is the bound $(u-\tfrac54)\zeta_K^{(2)}(4u-3)H_\beta(u,r) \ll r^{3/2-\sigma+\varepsilon}$ for the completed quartic Gauss series, with the analogous sextic bound used in Section 4; these control growth in the twist $md^2$ (quartic) or $md^4$ (sextic). Phragmén–Lindelöf interpolation between the lines $\mathrm{Re}\,u = 3/4+\eta$ and $3/2+\eta$, followed by Bochner's tube theorem and the theorem preserving bounds under tube continuation, produces the convex-hull regions whose boundaries at $w = 1/2$ are $\mathrm{Re}\,s = 49/58$ and $\mathrm{Re}\,s = 23/26$.
Extended reading notes
Core claim
The central claim is Theorem 1.1: assuming GRH for Dirichlet $L$-functions and for finite-order Hecke $L$-functions over $\mathbb{Q}(i)$ and $\mathbb{Q}(\sqrt{-3})$, the smoothed first moment over admissible characters of exact order $j$ equals $C_j\hat{\Phi}(1)Q + O(Q^{\theta_j+\varepsilon})$ with $\theta_4 = 49/58$, $\theta_6 = 23/26$, and positive constants $C_4$, $C_6$ given explicitly in (3.46) and (4.32). The proof continues the double Dirichlet series $A_j(s,w)$ to a region whose boundary at $w=1/2$ is exactly $\mathrm{Re}\,s = \theta_j$, then shifts the Mellin contour across the single pole at $s=1$. The enabling step is a joint estimate for the two sums left after Möbius inversion removes rational prime divisors from the Gaussian or Eisenstein modulus: with $\mu(d)$ retained, the pair factors as a quotient $L(z,\chi)/L^{(2)}(t,\chi^{2})$ times a convergent correction product in the quartic case, and similarly with $\chi^{4}$ in the sextic case, so GRH bounds numerator and reciprocal simultaneously. Combining this with the twist-exponent estimate for the completed quartic Gauss-sum series (Proposition 2.1) and the sextic analogue, then interpolating by Phragmén–Lindelöf with continuation to the convex hull via Bochner's tube theorem, gives boundary lines $58\,\mathrm{Re}\,s + 40\,\mathrm{Re}\,w = 69$ and $26\,\mathrm{Re}\,s + 20\,\mathrm{Re}\,w = 33$. The main terms are the same as in the earlier work, so the improvement is confined to the error term.
Load-bearing premise
The entire exponent $49/58$ rests on the cited bound that each completed quartic Gauss-sum series grows like $N(\nu)^{(3/2-\sigma)/2}$ with the factor $\zeta_K^{(2)}(4u-3)$ retained; if the true growth were worse, the region $58\,\mathrm{Re}\,s+40\,\mathrm{Re}\,w>69$ would shrink and the error would degrade, and the sextic case leans on the analogous growth bound from the cited sextic lemma.
Editorial extensions
If this is right
- Under GRH, both families gain power-saving first-moment error bounds: the exponents fall from $9/10$ and $13/14$ to $49/58$ and $23/26$, gains of $8/145$ and $4/91$, with the same main constants $C_4$, $C_6$.
- The error exponents come from the boundary lines $58\,\mathrm{Re}\,s+40\,\mathrm{Re}\,w=69$ and $26\,\mathrm{Re}\,s+20\,\mathrm{Re}\,w=33$, which at $w=1/2$ cut exactly at $\mathrm{Re}\,s=49/58$ and $\mathrm{Re}\,s=23/26$.
- The possible second poles at $s+w=5/4$ (quartic) and $s+w=7/6$ (sextic), meeting $w=1/2$ at $s=3/4$ and $s=2/3$, lie left of the new contours, so the asymptotics are power-saving but do not yet resolve secondary terms; the author states that these regions do not separate them.
- It suffices to assume GRH for the characters that actually occur in the proof and their powers, a hypothesis narrower than GRH for all Dirichlet and Hecke $L$-functions.
- The errors concern rational Dirichlet families averaged by conductor; the author notes they are not asserted to improve the corresponding Hecke-family exponents of the ideal-averaged families.
Reading between the lines
- My inference: the same retention of the Möbius function should transplant to other fixed-order families, such as cubic characters over $\mathbb{Q}(\sqrt{-3})$, whenever a twist-exponent bound for the Gauss-sum generating series is available; the pattern of the two gains suggests the payoff grows with the order of the characters.
- My inference: pushing the continuation across the lines $s+w=5/4$ and $s+w=7/6$ is the natural next target, with concrete rewards — secondary terms on the scale $Q^{3/4+\varepsilon}$ and $Q^{2/3+\varepsilon}$ at $w=1/2$ that the author explicitly leaves unresolved.
- My inference: because GRH controls the reciprocal $L$-functions appearing in the quotient identity, twisted first moments $\sum_\chi L(1/2,\chi)\chi(a)$ for these same families should admit the same power-saving treatment, giving a smoothing tool beyond the unweighted moment.
Original abstract
We study the smoothed first moments of primitive quartic and sextic Dirichlet $L$-functions for which every nontrivial power of the character remains primitive. Assuming the generalized Riemann hypothesis for the Dirichlet and finite order Hecke $L$-functions occurring in the proof, we obtain asymptotic formulas with errors $O(Q^{\frac{49}{58}+\eps})$ and $O(Q^{\frac{23}{26}+\eps})$, respectively. We follow the double Dirichlet series approach of Gao and Zhao. The main ingredient is a joint estimate for the two sums arising from the removal of rational prime divisors, with the M\"obius coefficients retained. This gives a quotient of Dirichlet $L$-functions. Under GRH, we bound both the Dirichlet $L$-functions and their reciprocals. Combining the joint estimate with estimates for the generating series of Gauss sums gives larger regions of meromorphic continuation. These regions do not separate the possible secondary terms.
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.
The abstract, sentence by sentence
We study the smoothed first moments of primitive quartic and sextic Dirichlet $L$-functions for which every nontrivial power of the character remains primitive. Assuming the generalized Riemann hypothesis for the Dirichlet and finite order Hecke $L$-functions occurring in the proof, we obtain asymptotic formulas with errors $O(Q^{\frac{49}{58}+\eps})$ and $O(Q^{\frac{23}{26}+\eps})$, respectively. We follow the double Dirichlet series approach of Gao and Zhao. The main ingredient is a joint estimate for the two sums arising from the removal of rational prime divisors, with the M\"obius coefficients retained. This gives a quotient of Dirichlet $L$-functions. Under GRH, we bound both the Dirichlet $L$-functions and their reciprocals. Combining the joint estimate with estimates for the generating series of Gauss sums gives larger regions of meromorphic continuation. These regions do not separate the possible secondary terms.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the smoothed first moments of primitive quartic and sextic Dirichlet $L$-functions for which every nontrivial power of the character remains primitive. | contribution 0.97 | 0.91 | 0.07 | asserted 0.86 |
| s1 | Assuming the generalized Riemann hypothesis for the Dirichlet and finite order Hecke $L$-functions occurring in the proof, we obtain asymptotic formulas with errors $O(Q^{\frac{49}{58}+\eps})$ and $O(Q^{\frac{23}{26}+\eps})$, respectively. | result 0.99 | 0.94 | 0.83 | asserted 0.96 |
| s2 | We follow the double Dirichlet series approach of Gao and Zhao. | method 1.00 | 0.57 | 0.02 | asserted 1.00 |
| s3 | The main ingredient is a joint estimate for the two sums arising from the removal of rational prime divisors, with the M\"obius coefficients retained. | method 0.61 | 0.84 | 0.06 | asserted 1.00 |
| s4 | This gives a quotient of Dirichlet $L$-functions. | method 0.36 | 0.74 | 0.03 | asserted 0.99 |
| s5 | Under GRH, we bound both the Dirichlet $L$-functions and their reciprocals. | method 0.93 | 0.92 | 0.04 | asserted 1.00 |
| s6 | Combining the joint estimate with estimates for the generating series of Gauss sums gives larger regions of meromorphic continuation. | result 0.42 | 0.89 | 0.06 | asserted 1.00 |
| s7 | These regions do not separate the possible secondary terms. | limitation 0.78 | 0.76 | 0.05 | asserted 0.99 |
What was examined, and what was not
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References
Discussion
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