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REVIEW 3 major objections 4 minor 45 references

On the $\beta=2$ Partition function for Dirichlet $L$-functions in the $q$-aspect

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For most Dirichlet characters modulo a large prime q, the maximum of |L(1/2+ih,chi)| over |h| <= 1/2 is bounded by log q/(log log q)^{3/4+o(1)}.

desk verdict Plausible new machinery for q-aspect partition function moments, but Theorem 3's final bound has a concrete arithmetic slip and key lemmas are asserted, so the headline maximum result is not established as written. read the letter →

arxiv 2608.09906 v1 pith:ECH3UBP5 submitted 2026-08-10 math.NT math.PR

Christopher Atherfold This is my paper · ORCID
classification math.NTmath.PR MSC 11M06
keywords DirichletL-functionsq-aspectbeta=2partitionfunctionFyodorov–Hiary–KeatingconjectureSaksman–WebbmesoscopicintervalsSteinhausrandommultiplicativefunctionsGaussianchaos
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

This paper sets out to control the size of Dirichlet $L$-functions at the central point for typical characters modulo a large prime $q$, through the $\beta=2$ partition function $Z_2(q,\chi,\theta)=\int_{|h|\le \log^\theta(q)/2}|L(1/2+ih,\chi)|^2\,dh$ with $\theta\in(-1/2,0]$. It proves moment upper bounds for this partition function, both at $\theta=0$ and on shrinking intervals, after conditioning on the frozen contribution of the $P_\theta$-smooth integers; the bounds carry the $\sqrt{\log\log q}$ Seneta--Heyde correction predicted by the Saksman--Webb conjecture. Its main theorem is a tail estimate: for $0\le U\le \log\log q$, the proportion of characters with $\max_{|h|\le 1/2}|L(1/2+ih,\chi)|\ge e^U\log q/(\log\log q)^{3/4}$ is bounded by a constant times $e^{-2U}(\log\log\log q)^2(\log\log\log q+U)$. If correct, this establishes the $q$-aspect analogue of the Fyodorov--Hiary--Keating upper bound and implies that all but $o(q)$ characters have maxima of size at most $\log q/(\log\log q)^{3/4+o(1)}$.

What carries the argument

The central object is the $\beta=2$ partition function $Z_2(q,\chi,\theta)$ together with its randomised version, where character sums are replaced by Steinhaus multiplicative functions. The carrying mechanism is Lemma 3, the randomisation estimate that equates character averages of smooth functions of short Dirichlet polynomials to averages over Steinhaus random multiplicative functions, up to a controllable error; this makes it possible to condition on the value of $Y_{P_\theta}(0)$, the short polynomial over $P_\theta$-smooth integers, without destroying orthogonality. Proposition 3.1 then converts the multipoint implicit conditioning into a single-point conditioning bound, using a chaining argument over the scale of the $P_\theta$-smooth contribution. For Theorem 3, the additional mechanism is a Cauchy integral formula that surrounds the point where the maximum is attained by a small rectangle, reducing the maximum to integrals of a factored Dirichlet polynomial that are estimated by fourth-moment and partition-function bounds under the barrier event $\widetilde G_\chi$.

What would settle it

One could test the central claim by computing, for a moderately large prime $q$ and some $\theta<0$, the conditioned expectation in Theorem 2 with $P=q^{1/(\log\log q)^8}$: if it exceeds a constant times $e^{2W}\log^{1+2\theta}(q)$ at $r=1$, Proposition 3.1 or Lemma 13 would be falsified. A second check would be to show, for some $U$ in the range, that the proportion of characters with maximum above $e^U\log(q)/(\log\log q)^{3/4}$ is $\gg e^{-2U}(\log\log\log q)^3$, which would contradict Theorem 3.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the randomisation scheme of [Har19] transfers the character-averaged problem to Steinhaus random multiplicative functions, where an explicit one-point conditioning replaces the multipoint barrier. For $\theta<0$ this yields, uniformly in $r\in[0,1]$ and $W\in\mathbb{R}$, the bound \[ \frac{1}{q-1}\sum_{\chi\bmod q}\left(Z_2(q,\chi,\$\theta$)^r \mid Y_{P_\$\theta$}(0)\in[W,W+1]\right)\ll \left(\frac{$e^{{2W}}$\$log^{{1+2\theta}}$(q)}{1+(1-r)\sqrt{\log\log q}}\right)^r, \] with the analogous unconditional bound at $\theta=0$. The same machinery, combined with a Cauchy-integral approximation of the maximum over $|h|\le 1/2$, gives Theorem 3's tail bound for the maximum itself. In short, the paper claims the $\beta=2$ partition function is the right object for extracting the full ballot-theorem savings that determine the typical maximum of Dirichlet $L$-functions in the $q$-aspect.

Load-bearing premise

The load-bearing premise is that the three technical lemmas controlling conditioned random Euler products, Lemmas 13--15, remain valid at the paper's smoothing parameter $P=q^{1/(\log\log q)^8}$; if any of them fails there, the moment bounds and the maximum tail estimate do not follow.

Editorial extensions

If this is right

  • If Theorem 3 is correct, at least $q(1-o(1))$ characters modulo $q$ satisfy $\max_{|h|\le 1/2}|L(1/2+ih,\chi)|\ll \log(q)/(\log\log q)^{3/4+o(1)}$.
  • For any $U(q)\to\infty$, the number of characters whose maximum reaches $e^{U(q)}\log(q)(\log\log\log q)^{3/2}/(\log\log q)^{3/4}$ is $o(q)$.
  • The tail bounds of Corollary 1 imply that the normalisation $\log(q)\sqrt{\log\log(q)}$ is of the correct order for the $\beta=2$ partition function, supporting the conjectured Gaussian multiplicative chaos limit.
  • The paper states that the same argument generalises to the $t$-aspect for $\zeta(1/2+it)$ on mesoscopic intervals, giving improved bounds in the range $\theta\in(-1/2,0)$.

Reading between the lines

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

  • If the upper bound in Theorem 3 is sharp, a matching lower bound of order $e^{-2U}$ should be within reach of the probabilistic arguments used in the $t$-aspect literature, but the paper does not attempt that step.
  • The explicit conditioning on $Y_{P_\theta}(0)$ suggests a general recipe for mesoscopic problems: condition first on the frozen smooth contribution, then treat the log-correlated part by Gaussian-walk barrier events; this recipe could be tested on short-interval character sums or random matrix models.
  • Because Lemmas 13--15 are imported without full proofs, a reader who wants to verify Theorem 3 independently should check those lemmas at the boundary parameter $P=q^{1/(\log\log q)^8}$; failure there would tighten the admissible range of $P$ and $\varepsilon$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops Harper's randomisation method for the q-aspect beta=2 partition function ∫_{|h|≤log^θ(q)/2}|L(1/2+ih,χ)|^2 dh. It proves moment bounds (Theorems 1 and 2) with explicit conditioning on a smooth-number Dirichlet polynomial, and uses a Cauchy-integral argument to derive a tail bound (Theorem 3) for characters whose maximum on |h|≤1/2 exceeds e^U log q/(log log q)^{3/4}. The advertised consequence is that all but o(q) characters have max_{|h|≤1/2}|L(1/2+ih,χ)| ≪ log q/(log log q)^{3/4+o(1)}. The proof is unconditional and has no fitted parameters, but it depends on several unproved lemmas imported from Harper's framework.

Significance. If the proofs were complete, the paper would make a substantial contribution: Theorem 3 would give the q-aspect analogue of the Fyodorov–Hiary–Keating upper tail up to second order, and Theorems 1–2 would confirm the Saksman–Webb normalisation for mesoscopic intervals. The paper's assets are its explicit, parameter-free conditioning, the chaining argument of Proposition 3.1, and the clean reduction of the maximum problem to a conditioned Euler-product estimate. However, the final counting step contains a concrete arithmetic error, and the key lemmas underpinning Propositions 4.2 and 5.2 are stated without proof. The qualitative o(q) application may survive a weaker form of Theorem 3, but the paper's central precision claim is not established.

major comments (3)
  1. [§5, proof of Theorem 3 (after (5.4))] The passage from Proposition 5.2 to the final bound is arithmetically incorrect. With V=e^{-U}(log log q)^6 one has log V=-U+6 log log log q, so the second min-factor in Proposition 5.2 has argument (U+log log log P+log V)/sqrt(log log P)=7 log log log q/sqrt(log log q)=o(1) and is therefore identically 1. Hence Proposition 5.2 gives N≪q e^{-2U}(log log q)^{3/2} min{1,(U+log log log q)/sqrt(log log q)}, and for 0≤U≤log log log q this is q e^{-2U}(log log q)(U+log log log q), not the claimed q e^{-2U}(U+log log log q)(log log log q)^2. The discrepancy is at least a factor (log log q)/(log log log q)^2, and it is larger when U is of size sqrt(log log q). Thus the stated uniform bound in Theorem 3 is not established by the written proof.
  2. [§5, Eqs. (5.3)–(5.4) and §5.1, Lemma 15] There is a parameter-balancing obstruction in addition to the arithmetic slip. To make the (5.3) contribution small enough, V must be a large positive power of log log q; with the chosen V=e^{-U}(log log q)^6 the second min-factor in Proposition 5.2 becomes 1 and the (5.4) contribution carries an extra factor (log log q)^{3/2}. If instead one tries to make that min-factor of size log log log q/sqrt(log log q), which is what the claimed bound would require, one must take V≈e^{-U}, and then the (5.3) contribution becomes too large. No choice of V in the displayed argument balances both terms; the proof needs a new mechanism, not merely a change of constants.
  3. [§4.1 (Lemma 13) and §5.1 (Lemmas 14–15)] The paper's central estimates rely on Lemmas 13, 14 and 15, none of which is proved. Lemma 15, in particular, is the only input that converts the conditioned character average in Proposition 5.2 into the claimed min-factor bound, and the paper states only that the proof is 'rather tricky' and follows Harper's arguments. Since these lemmas are needed in parameter regimes not literally covered by the cited statements, for example P=q^{1/(log log q)^8}, V=e^{-U}(log log q)^6, and ε=(log log q)^{-2} in Theorem 3, the conditional probabilities and moment bounds on which the main theorem rests cannot be verified from the manuscript. This is a load-bearing gap, not a presentation issue.
minor comments (4)
  1. [Lemma 10 proof] In the first displayed term after the triangle inequality, the expectation and absolute values are missing; it should read E|Σ_{m>q^{εθ}, P^+(m)≤Pθ} f(m)/m^{1/2+ih}|^2.
  2. [Proposition 4.2] In the displayed formula, the differential dh appears inside the expectation; it should be E_char[1_{Gχ}|...|^2|...|^2] dh, with dh outside.
  3. [Section 1.3] The sentence claiming that Theorem 3 'could be further improved by a more delicate handling of the Dirichlet polynomial over the rough numbers' sits oddly with the fact that the stated bound is not currently established; the discussion should be revised after the proof is repaired.
  4. [Section 4] The quantity P_q(θ) is used before it is explicitly defined; define it at its first occurrence for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof uses external Harper blackboxes and universal constants, and no fitted parameter is renamed as a prediction.

full rationale

The paper's derivation chain is a standard upper-bound argument: approximate the L-function by Dirichlet polynomials, transfer character averages to Steinhaus random multiplicative functions via Harper's randomisation (Lemma 3, cited from [Har23]), apply probabilistic barrier estimates (Lemmas 8-9, cited from [Har19]/[Har20], and Lemmas 13-15, stated with proof outlines and attributed to Harper), then count exceptional characters. None of these steps assumes the theorem being proved, and no parameter is fitted to the target quantity. The constants are absolute, and the bounds are claimed uniformly in q and U. The event definitions (e.g., eG_chi and H_chi(h;V)) are motivated by the conjectures, but they are used as proof devices, not as equivalent reformulations of the conclusions. The heavy lemmas are imported from Harper's independent work, not from the author's own prior results, so the self-citation concern does not arise; the paper even explicitly says for Lemma 13, 'While we will not prove the above, we will outline how the proof works,' and for Lemma 14, 'Again, this goes through without change to the analogous result Lemma 6 in [Har19].' This is transparent reliance on external machinery. A separate arithmetic discrepancy exists in the final counting step of Theorem 3: after (5.4), substituting V=e^{-U}(log log q)^6 makes the second min-factor in Proposition 5.2 identically 1, leaving roughly q e^{-2U}(log log q)^{3/2} min{1,(U+LLL(q))/sqrt(LL(q))}, which does not obviously imply the stated q e^{-2U}(U+LLL(q))(LLL(q))^2. That is a correctness concern, not circularity, and it does not change the circularity score.

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

The central bounds rest on Harper's randomisation and Gaussian comparison machinery, and on the paper's own unproved lemmas 13-15. No parameters are fitted to data; the constants are chosen for the proofs to satisfy the hypotheses of the quoted lemmas.

free parameters (4)
  • P (cutoff for smooth/rough split) = q^{1/(log log q)^8}
    Chosen in the proofs of Theorems 1-3 to satisfy the hypotheses of Lemma 3; the quality of the moment bounds depends on this choice.
  • delta (smoothing parameter in partition of unity) = 1/(log log P)^2
    Chosen to balance the error terms in Lemma 1 and Lemma 3; smaller delta would worsen the error term.
  • V (threshold for large Dirichlet polynomial in Theorem 3) = e^{-U} (log log q)^6
    Chosen so that the event H_chi(h;V) being false is meaningful; the final bound depends on log(V).
  • epsilon (length of the smooth part) = 1/(log log q)^2 (plus epsilon_theta in mesoscopic case)
    Chosen with P to make the approximations in Lemmas 5 and 6 valid and to satisfy Lemma 3's condition.
assumptions (4)
  • standard math Approximate functional equation for Dirichlet L-functions (Lemma 4)
    Used to approximate L(1/2+ih,chi) by a Dirichlet polynomial of length sqrt(q)/2pi in the proofs of all theorems.
  • domain assumption Harper's randomisation estimate (Proposition 1 in [Har23], Lemma 3 here)
    Transfers character averages to Steinhaus random multiplicative function expectations; this is the central bridge of the paper.
  • domain assumption Probability results for Gaussian walks (Lemmas 8 and 9 and Harper's Lemmas 4-5 in [Har20])
    Used to show the barrier event G_chi holds with high probability; the adaptation to random Euler products is not proved in this paper.
  • ad hoc to paper The paper's Lemmas 13, 14, and 15
    These control conditioned random Euler products and are essential to Propositions 4.2 and 5.2, but are stated without proof or with only an outline.

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Pith. "Pith review of On the $\beta=2$ Partition function for Dirichlet $L$-functions in the $q$-aspect." pith.science (2026). https://pith.science/paper/ECH3UBP5

@misc{pith2026260809906,
  author       = {Pith},
  title        = {Pith review of: On the $\beta=2$ Partition function for Dirichlet $L$-functions in the $q$-aspect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ECH3UBP5}},
  note         = {Machine review of arXiv:2608.09906}
}
abstract

We study the $\beta=2$ partition function $\int_{|h| \leq \log^{\theta}(q)/2}|L(1/2+ih,\chi)|^2dh$ for typical Dirichlet characters modulo a large prime $q$ and $\theta \in (-1/2,0]$ motivated by a $q$-analogue of the Saksman--Webb conjectures. When $\theta <0$, we use Harper's randomisation argument to introduce explicit conditioning to recover moment upper bounds consistent with critical normalisation predicted there. As an application, we prove that for $q(1-o(1))$ Dirichlet characters modulo $q$, $\max_{|h| \leq 1/2}|L(1/2+ih,\chi)|\ll \frac{\log(q)}{(\log\log(q))^{3/4+o(1)}}$, establishing an upper bound matching the predictions of the $q$-analogue of the Fyodorov--Hiary--Keating conjectures up to second order.

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