{"id":"92e0f520-29ce-4f89-854c-e94ba28c6594","arxiv_id":"2608.09906","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Upper bounds for the q-aspect beta=2 partition function of Dirichlet L-functions and for the typical maximum, matching FHK predictions to second order.","lead":"The paper proves upper bounds for moments of the beta=2 partition function of Dirichlet L-functions in the q-aspect, and shows that almost all characters have maximum value at most log(q)/(log log q)^{3/4+o(1)}. This is the first matching upper bound for the q-analogue of the Fyodorov-Hiary-Keating conjecture.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3's final estimate has an arithmetic slip: with V=e^{-U}(log log q)^6, Proposition 5.2 gives an extra factor of log log q (or worse), so the stated e^{-2U}(log log log q)^2(log log log q+U) bound does not follow.","rationale":"The reader's weakest_assumption focused on Lemmas 13–15 being stated without complete proofs and adapted from Harper's arguments. That is a legitimate concern about self-containedness. However, the stress-test found a more immediate obstruction that does not depend on whether those lemmas are true: the final counting argument in Theorem 3 mis-simplifies the bounds supplied by Proposition 5.2. Setting V = e^{-U}(log log q)^6 makes the second min-factor equal to 1, leaving a factor (log log q)^{3/2} that is not cancelled by the claimed final bound. The stated theorem is stronger than what the proof yields; the discrepancy is a polylogarithmic factor, so the paper's broad qualitative picture may survive a correction, but the central theorem as stated is not proven by the present argument. For that reason the appropriate assessment is CONDITIONAL: the argument needs a corrected final estimate or revised parameter choices before the main advertised result can be accepted. The reader's identifying the missing lemma proofs as the weakest point is reasonable, but the concrete arithmetic slip is more load-bearing and should be addressed first.","tokens_in":29174,"tokens_out":8738,"duration_ms":82486,"concrete_test":"Independently re-derive the final display in the proof of Theorem 3: substitute V = e^{-U}(log log q)^6 into the two min-factors of Proposition 5.2, simplify the second min-factor to 1, and compare the resulting expression with the claimed bound q e^{-2U}(U+LLL(q))(LLL(q))^2. If the simplified expression contains a factor (log log q)(U+LLL(q)) (or (log log q)^{3/2}) instead of (U+LLL(q))(LLL(q))^2, the theorem's stated uniform tail is unsupported. This is a direct algebraic check requiring no numerical computation beyond asymptotic simplification.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 3 makes an unsupported simplification in its final counting step. After equation (5.4), Proposition 5.2 is quoted as giving the number of exceptional characters as\n\nN ≪ q (log log q)^{3/2} e^{-2U} min{1, (U+LLL(P))/sqrt(LL(P))} * min{1, (U+LLL(P)+log V)/sqrt(LL(P))}^2,\n\nwhere LLL = log log log. The paper sets V = e^{-U} (log log q)^6, so log V = -U + 6 LLL(q). Hence the second min-factor has argument (U + LLL(q) - U + 6 LLL(q))/sqrt(LL(q)) = 7 LLL(q)/sqrt(LL(q)) = o(1), and the second min-factor is identically 1 for large q. The first min-factor is min{1, (U+LLL(q))/sqrt(LL(q))}.\n\nThus the displayed bound is at least\n\nq e^{-2U} (log log q)^{3/2} min{1, (U+LLL(q))/sqrt(LL(q))}.\n\nIf U+LLL(q) ≤ sqrt(LL(q)), this equals q e^{-2U} (log log q)(U+LLL(q)); if U+LLL(q) > sqrt(LL(q)), it equals q e^{-2U} (log log q)^{3/2}. Neither is bounded by the claimed q e^{-2U}(U+LLL(q))(LLL(q))^2 for the full range 0 ≤ U ≤ log log q. The discrepancy is at least a factor of (log log q)/(LLL(q))^2, and it is larger when U is of order sqrt(log log q).\n\nThis is a concrete arithmetic error in the proof of the paper's central theorem: even granting Lemmas 13–15 (whose proofs are also omitted), the stated uniform tail bound does not follow from the written chain. The qualitative o(q) consequence may survive, since the extra factor is only polylogarithmic in log q, but the precise second-order bound in Theorem 3, which is the main advertised result, is not established as written.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":29692,"tokens_out":24778,"duration_ms":213752,"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":[{"comment":"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.","section":"§5, proof of Theorem 3 (after (5.4))"},{"comment":"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.","section":"§5, Eqs. (5.3)–(5.4) and §5.1, Lemma 15"},{"comment":"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.","section":"§4.1 (Lemma 13) and §5.1 (Lemmas 14–15)"}],"minor_comments":[{"comment":"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.","section":"Lemma 10 proof"},{"comment":"In the displayed formula, the differential dh appears inside the expectation; it should be E_char[1_{Gχ}|...|^2|...|^2] dh, with dh outside.","section":"Proposition 4.2"},{"comment":"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.","section":"Section 1.3"},{"comment":"The quantity P_q(θ) is used before it is explicitly defined; define it at its first occurrence for clarity.","section":"Section 4"}],"recommendation":"major_revision","confidential_remarks":"The reader's stress-test concern about the final counting step is accurate; I found no way around it with the choices made in §5. The paper is not suitable for publication in its present form, but the general approach has merit and Theorems 1–2 could justify a revision if the missing lemmas are supplied and the final bound is either corrected or stated in a weaker form that still supports the advertised application."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things. First, the paper genuinely extends Harper's randomisation to the q-aspect and proves upper bounds for moments of the beta=2 partition function, including a mesoscopic conditioned version, that look new and likely correct. Second, the proof of Theorem 3, the headline FHK-type maximum bound, does not close: there is an arithmetic slip in the final counting step, and the key probabilistic lemmas (13-15 and Harper's Lemmas 4-5) are quoted or outlined rather than proved.\n\nSpecifically, after (5.4) the paper quotes Proposition 5.2 as giving the tail count with the min factors. With V=e^{-U}(log log q)^6, the second min is 7 log log log q / sqrt(log log q) = o(1), hence equals 1. The first min then leaves either (log log q)(U+LLL(q)) or (log log q)^{3/2}, neither of which is bounded by the claimed (U+LLL(q))(LLL(q))^2. So the uniform tail bound in Theorem 3 is not derived from the displayed chain. The qualitative o(q) estimate may survive, but the second-order precision advertised is not supported.\n\nCredit where due: the mesoscopic explicit conditioning (Theorem 2 and Proposition 3.1) is a real idea, and the transfer machinery is used carefully. The paper is honest about its black boxes; it flags the omitted proofs. But being honest about what is missing is not the same as providing it.\n\nThe result matters. A correct q-aspect FHK upper bound at this precision would be a solid step. As it stands, the paper needs a serious revision: the final estimate in Theorem 3 must be reworked (the extra log log q factor may force a different choice of V or a sharper use of Lemma 15), and the omitted lemmas need to be supplied or precisely referenced with conditions checked. The parameter choices P and epsilon are plausible, but we cannot verify from the text.\n\nWho is this for? Specialists in multiplicative chaos and L-function maxima. It deserves a serious referee rather than a desk reject, because the approach is promising and Theorems 1-2 may stand independently. But I would not cite Theorem 3 as it stands. Send it to a referee who knows Harper's work, with instructions to check the arithmetic in the proof of Theorem 3 first.","headline":"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.","tokens_in":30190,"tokens_out":2327,"would_cite":false,"duration_ms":20022,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"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)}.","keywords":["Dirichlet L-functions","q-aspect","beta=2 partition function","Fyodorov–Hiary–Keating conjecture","Saksman–Webb conjecture","mesoscopic intervals","Steinhaus random multiplicative functions","Gaussian multiplicative chaos"],"falsifier":"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.","tokens_in":28963,"feed_emoji":"📈","tokens_out":15297,"duration_ms":114129,"temperature":0.7,"pith_summary":"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 $\\log q/(\\log\\log q)^{3/4+o(1)}$.","feed_headline":"Most Dirichlet L-functions top out at log q/(log log q)^{3/4}","feed_subtitle":"A new upper bound matches the q-aspect Fyodorov–Hiary–Keating prediction up to second order.","key_machinery":"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 $\\widetilde G_\\chi$.","core_discovery":"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\n\\[\n\\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,\n\\]\nwith 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.","pith_inferences":["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 $\\varepsilon$."],"forward_implications":["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 $\\theta\\in(-1/2,0)$."],"supporting_citations":[{"why":"Supplies the randomisation estimate that replaces character sums by Steinhaus random multiplicative functions, the key transfer used throughout.","marker":"[Har23]"},{"why":"Provides the random Euler product lemmas and the ballot-theorem savings that control the log-correlated part under a barrier event.","marker":"[Har19]"},{"why":"States the Gaussian-walk probability results that underpin the conditioned Euler product bounds in the paper.","marker":"[Har20]"},{"why":"Gives the Dirichlet polynomial discretisation lemma used to reduce maxima over $h$ to pointwise estimates.","marker":"[ABR20]"},{"why":"Supplies the chaining argument for mesoscopic intervals and the single-point conditioning framework used in Proposition 3.1.","marker":"[AH24]"},{"why":"Provides the approximate functional equation for Dirichlet $L$-functions used in every reduction to finite Dirichlet polynomials.","marker":"[DW65]"},{"why":"Proposes the conjectured maximum distribution whose $q$-aspect upper bound Theorem 3 establishes.","marker":"[FHK12]"},{"why":"Conjectures the Gaussian multiplicative chaos limit that fixes the normalisation of the partition function.","marker":"[SW20]"}],"fun_headline_variants":["New bound for Dirichlet L-function maxima matches predictions","Randomization proves tight max for almost all Dirichlet L-functions","Partition function approach pins down Dirichlet L-function maxima","Almost all Dirichlet L-functions hit log q over (log log q)^(3/4)","Matches q-aspect FHK prediction for typical Dirichlet characters"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["New bound for Dirichlet L-function maxima matches predictions","Randomization proves tight max for almost all Dirichlet L-functions","Partition function approach pins down Dirichlet L-function maxima","Almost all Dirichlet L-functions hit log q over (log log q)^(3/4)","Matches q-aspect FHK prediction for typical Dirichlet characters"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002016,"raw_usage":{"total_tokens":7887,"prompt_tokens":1001,"completion_tokens":6886,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":6794}},"tokens_in":617,"tokens_out":6886,"duration_ms":45510,"temperature":1.0,"reasoning_tokens":6794,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:41:12.944941+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}