REVIEW
Large values of cubic and quartic Dirichlet $L$-functions of prime moduli
Number Theory · Submitted 2 Sep 2026 · 22 references
Pith's verdict: Extends the Bondarenko-Seip resonator to cubic and quartic prime-modulus families under GRH; the cubic half rests on published work, but the quartic half pivots on an unverified same-author preprint—needs closing before publication.
T0 · AI review
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Referee report
The paper
The paper claims
Exponential lower bounds for cubic and quartic Dirichlet L-functions at the central point, under GRH.
The reading
What carries the argument
The central object is the long resonator, a Dirichlet polynomial $R_\chi = \sum_{m\in R} \psi(m)\chi(m)$ with coefficients $\psi$ defined in (3.1) to concentrate weight on integers composed of many small primes. The argument uses the approximate functional equation, smoothed character sums (Lemma 2.8 under GRH), and the key bound (Proposition 2.4) on the twisted sums $H_Z(r;\psi)$ of Gauss sums, which for $j=3$ gives exponent $4/5$ and for $j=4$ gives exponent $7/8$.
Extended reading notes
Core claim
Theorem 1.1 states that, assuming GRH, for sufficiently large $X$ there exist primes $p$ with $X<p\leq 2X$ and primitive cubic (resp. quartic) characters $\chi$ mod $p$ such that $|L(1/2,\chi)|$ is at least the exponential quantity given above. The constant $\sqrt{2}/\pi$ for cubic and $\sqrt{3/2}/\pi$ for quartic arise from the exponents $4/5$ and $7/8$ in the bound of Proposition 2.4, together with an optimisation of the resonator parameters $A$ and $B$.
Load-bearing premise
The quartic part of Theorem 1.1 depends on Proposition 2.4 for $j=4$, which is taken from an unpublished preprint [14] and has not yet passed peer review; if that bound turns out to be false, the quartic lower bound would be unsupported.
Editorial extensions
If this is right
- The lower bounds show that cubic and quartic $L$-functions have extreme value growth comparable to that already known for quadratic twists, though with different constants.
- The method suggests that similar exponential lower bounds should hold for other families of $L$-functions with suitable Gauss-sum relations.
- Given GRH, the bounds are essentially the best that can be obtained from the current long-resonator framework; improving the exponents in Proposition 2.4 would sharpen the constants.
Reading between the lines
- The quartic result rests on an unpublished bound (Proposition 2.4 for $j=4$ from [14]), so its validity is conditional on that preprint being correct. This is a weaker link than the cubic case, which uses a published theorem.
- It is plausible that the constants $\sqrt{2}/\pi$ and $\sqrt{3/2}/\pi$ are not optimal; a finer analysis of the resonator construction might improve them.
- One could test the robustness of the quartic bound by attempting to derive a similar bound from published techniques, or by checking specific numerical examples for small $p$.
Original abstract
In this paper, we apply the resonator method to exhibit large values at the central point of the families of cubic and quartic Dirichlet $L$-functions of prime moduli under the generalized Riemann hypothesis.
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
In this paper, we apply the resonator method to exhibit large values at the central point of the families of cubic and quartic Dirichlet $L$-functions of prime moduli under the generalized Riemann hypothesis.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we apply the resonator method to exhibit large values at the central point of the families of cubic and quartic Dirichlet $L$-functions of prime moduli under the generalized Riemann hypothesis. | contribution 0.94 | 0.94 | 0.88 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
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References
Discussion
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