{"id":"0a6fa3f5-24db-47d2-a599-756dc3b8e0aa","arxiv_id":"2607.20041","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Riemann zeta-function is shown to have the universality and Bohr–Jessen distribution properties under logarithmic-power shifts (log t)^α, α>1, with quantitative discrepancy bounds.","lead":"This paper proves new distribution results for the Riemann zeta-function when the usual vertical shift it is replaced by slower or faster functions γ(t), including logarithmic powers. It obtains discrepancy bounds for the Bohr–Jessen limit theorem and a universality theorem for shifts (log t)^α that earlier methods could not handle.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 6.5's approximation-error bound is false as written: the chosen Y is too small, leaving a repairable gap in the proof of Theorem 1.5.","rationale":"The reader's weakest assumption concerned the α>1 restriction and the zero-density threshold x_Φ(α1). My load-bearing concern is a different, more concrete issue: the explicit Y in Lemma 6.5 is too small for Lemma 4.1's Dirichlet-polynomial approximation to have vanishing error. This affects the proof of Proposition 6.1, which is the limit theorem behind Theorem 1.5. I found the flaw by plugging the stated Y into Lemma 4.1 with the correct shifted interval endpoint (log T)^{α_j}; the error is O(log log T), not O((log log T)^{-1}) as claimed. Because a larger polynomial-in-loglogT choice of Y repairs the estimate without breaking the neighboring lemmas, the central claim is not disproved; the paper should remain under the reader's conditional verdict pending this correction. Hence I recommend no change to the verdict, and I mark agreement as disagree since the reader's stated weakest assumption did not identify this point.","tokens_in":19872,"tokens_out":27013,"duration_ms":239871,"concrete_test":"Recompute the first-term bound in Lemma 6.5 by substituting y=(loglogT)^{4/(σ0−1/2)} and T=(logT)^{α_j} into Lemma 4.1's error term; if the result is of size loglogT rather than (loglogT)^{-1}, the proof as stated fails. Then re-run the same estimate with Y=(loglogT)^N, N>8/(σ0−1/2), and check that the Lemma 4.1 error, the Lemma 6.3 error term, and the Lemma 6.2 exceptional measure all vanish.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Lemma 6.5 (Section 6), Y is set to (log log T)^{4/(σ0−1/2)}. The first term bounds ∫_{X'_K(T)} |logζ(σ+i(logt)^{α_j}) − Σ_{2≤n≤Y} Λ(n)n^{−σ−i(logt)^{α_j}}/log n| dt by Lemma 4.1. There, T is the endpoint of the interval for the zeta argument, i.e. T=(logT)^{α_j}, so log T in the error is O(log log T). Lemma 4.1 gives a pointwise error O(Y^{(1/2−σ0)/2}(log log T)^3) = O((log log T)^{−2}(log log T)^3) = O(log log T). This does not tend to 0 and is not the displayed O((log log T)^{−1}). Consequently the L1-approximation logζ≈logζ_X on X_K(T) is not established, and Proposition 6.1 (hence Theorem 1.5) has a genuine proof gap. The gap is concrete, not fatal: taking Y=(log log T)^N with N>8/(σ0−1/2) makes the approximation error tend to 0 while Lemma 6.2 and Lemma 6.3 errors remain o(1).","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the value distribution of the Riemann zeta-function under general shifts. Theorem 1.4 gives a discrepancy bound D_{σ,γ}(T) ≪ (log γ(T))^{-σ} for shifts in a class F′. Theorem 1.5 proves joint universality for logarithmic-power shifts (log τ)^{α_j}, 1<α_1<...<α_r, in a strip depending on the zero-density exponent Φ, and under the Riemann hypothesis extends the strip to 1/2<σ<1. Theorem 1.6 gives a quantitative Bohr–Jessen discrepancy D_{σ,r}(T) ≪ ((r−1) log log T)^{-σ} for the single shift (log t)^r. The proofs combine zero-density estimates, Dirichlet polynomial approximations, Beurling–Selberg smoothing, and the probabilistic Bagchi–Matsumoto framework. The main weaknesses are a false estimate in Lemma 6.5 and the largely deferred proofs of the key statements in Section 7.2.","tokens_in":20252,"tokens_out":28197,"duration_ms":239202,"significance":"If the results are correct, the paper makes a genuine contribution: it unifies discrepancy estimates for a broad class of shifts (Theorem 1.4) and, more strikingly, establishes joint universality and quantitative value-distribution results for logarithmic-power shifts, which are outside the polynomial/exponential coverage of previous work. The methods are mostly standard and the use of zero-density estimates is principled; there are no fitted parameters. The paper ships no code or machine-checked proofs, but the probabilistic framework is reproducible from the references. However, the validity of the central universality claim currently hinges on a false estimate in Lemma 6.5, and the main quantitative theorem (Theorem 1.6) rests on unproved 'repetition' arguments. These are local and repairable, but they must be fixed before the results can be accepted.","major_comments":[{"comment":"The displayed estimate '≪(log log T)^{-1}' in the proof of Lemma 6.5 is false for the chosen Y=(log log T)^{4/(σ0−1/2)}. Applying Lemma 4.1 with the endpoint of the zeta interval T_ζ=(log T)^{α_j} gives a pointwise error O(Y^{(1/2−σ0)/2}(log T_ζ)^3)=O((log log T)^{-2}(log log T)^3)=O(log log T), not o(1). Consequently the L^1 approximation of logζ by logζ_X on X_K(T) is not established, and Proposition 6.1, hence Theorem 1.5, lacks a valid proof. The gap is concrete and fixable: taking Y=(log log T)^N with N>8/(σ0−1/2) makes the first error tend to 0 while the errors in Lemmas 6.2 and 6.3 remain o(1).","section":"§6, Lemma 6.5"},{"comment":"The proof of Theorem 1.6 rests on Lemma 7.2 and Proposition 7.3, but neither is actually proved. Lemma 7.2 is dismissed with 'we can prove this lemma in the same way as Lemma 4.3', and Proposition 7.3 with 'the proposition follows by repeating the proof of Proposition 4.5'. These are the decisive estimates that produce the (r−1) log log T discrepancy. Because γ(t)=(log t)^r is not in the class F (it fails (F3)), the repetition is not completely formal; the parameter choices and error terms must be checked. The authors should supply the full arguments, in particular the verification that the analogue of (4.2) holds with error O((log log T)^{-5}).","section":"§7.2, Lemma 7.2 and Proposition 7.3"},{"comment":"The proof sets L=log log log T at the start, but then concludes D_{σ,r,Y}(T)≪(log log T)^{-2}+L^{-1}≪((r−1)log log T)^{-σ}. For L=log log log T, L^{-1}=1/log log log T, which is not O((log log T)^{-σ}) for any σ>0. The final inequality only follows if L is the quantity c_3((r−1)log log T)^σ from Proposition 7.3. As printed, the derivation of the stated discrepancy bound is not justified. Please correct the definition of L and the associated display.","section":"§7.2, proof of Theorem 1.6"}],"minor_comments":[{"comment":"The final exponent '1/2−σ0' in Lemma 4.2 does not follow from the cited zero-density estimate [28, Theorem 9.19.A] and the preceding display; for σ0>1/2 it would imply N(σ0,T) decays, which is impossible. The exponent appears to be a typo; please correct it to the value actually obtained from the zero-density theorem and condition (F4).","section":"§4, Lemma 4.2"},{"comment":"The symbol r is used both as the number of joint components in Theorem 1.5 and as the exponent in Theorem 1.6. This is confusing, especially in Section 7 where both roles occur; please use a different symbol, e.g. ν, for one of them.","section":"Notation"},{"comment":"The notation ℓ([...]; σ0, Y) is used in the proof of Lemma 6.5, but Lemma 4.1 defines ℓ only with a single parameter y. Please align the definitions so that the role of σ0 and Y is clear.","section":"§6, Lemma 6.5"},{"comment":"The proof states 'Note that we use (F4) to evaluate ...' but the verification is not shown. Since condition (F4) is a new structural assumption, a few lines explaining the estimate would be helpful.","section":"§4, Proposition 4.5"},{"comment":"The phrase 'logarithmic-power shifts' should specify that α>1 is assumed; the case α=1 is explicitly out of reach and is only discussed in §8.","section":"Abstract/Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper is potentially publishable after substantial revision. The main results are plausible and interesting, but the proof of the central new theorem (Theorem 1.5) has a concrete false estimate in Lemma 6.5, and Theorem 1.6 relies on deferred proofs in Section 7.2. The issues are local and repairable, so I do not see grounds for rejection, provided the authors supply the missing details. Editors may want to send the revision back to a referee familiar with the Bagchi–Matsumoto machinery."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper does something new — it treats (log t)^α shifts with α>1, which fall outside the author's earlier class F and outside Pánkowski-type results. The universality theorem (Theorem 1.5) and the quantitative Bohr–Jessen bound (Theorem 1.6) are plausible and well-motivated. Sections 3–6 show genuine command of the Bagchi/Matsumoto/LLR machinery; Lemma 6.3's mean-value estimate and Lemma 6.2's exceptional-set preimage estimate are the real new pieces.\n\nThere is, however, a concrete gap in Lemma 6.5. With Y=(log log T)^{4/(σ0−1/2)}, applying Lemma 4.1 on the interval (log t)^{α_j} gives pointwise error O(Y^{(1/2−σ0)/2}(log log T)^3)=O(log log T), not the displayed O((log log T)^{-1}). So the L1 approximation logζ≈logζ_X on X_K(T) is not established as written. This is repairable — take Y=(log log T)^N with N>8/(σ0−1/2); Lemma 6.2 and Lemma 6.3 errors remain o(1) — but it is a real proof gap, not a typo. The stress-test note is right.\n\nSecond, Theorem 1.6 is incomplete as it stands: Lemma 7.2 and Proposition 7.3 are delegated to “by repeating” the Section 4 arguments, and those are the decisive steps. A referee should ask for the details. Also, the claim that logarithmic-power shifts are not covered by previous results should be pinned down against Pánkowski's hypotheses; his τ^a(log τ)^b appear to have a>0, so the log-power case is indeed new, but the α=1 boundary is explicitly out of reach and should be stated as such.\n\nThe extracted text has garbled formulas, so some estimates could not be checked line-by-line; that is partly an artifact, but a clean manuscript is needed. The innovation is not routine — the log-power shift family is genuinely outside the earlier polynomial-growth class — and the main theorems are likely correct. Someone in value distribution will want to cite this once the gap is fixed.\n\nFor a referee: send it, with a request to fix Lemma 6.5 and supply the Section 7.2 details. Not a desk reject, but not accept-as-is.","headline":"Genuinely new results for log-power shifts, with a concrete (repairable) approximation gap in Lemma 6.5 and an underproved Theorem 1.6; deserves peer review.","tokens_in":20671,"tokens_out":4245,"would_cite":true,"duration_ms":39096,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26","11M41","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the Riemann zeta-function is jointly universal under shifts by powers of log t, and gives explicit discrepancy rates for the Bohr-Jessen limit theorem under very general shifts.","keywords":["Riemann zeta-function","universality","Bohr-Jessen limit theorem","discrepancy","logarithmic shifts","value distribution","zero-density estimates","Dirichlet polynomials"],"falsifier":"Find one rectangle R and one sequence T_k for which |P_{T_k}(log ζ(σ+i(log t)^r)∈R)-P(log ζ(σ,X)∈R)| grows faster than ((r-1) log log T_k)^{-σ}, with r>1 and σ>x_Φ(r); that would disprove the main quantitative theorem.","tokens_in":19804,"feed_emoji":"","tokens_out":3870,"duration_ms":33905,"temperature":0.7,"pith_summary":"The paper is trying to establish how much of the zeta-function's value distribution survives when the usual vertical shift it is replaced by a slowly growing shift iγ(t). For a broad class of fast-growing shifts—polynomial, exponential, and beyond—it proves a quantitative Bohr-Jessen discrepancy bound of order (log γ(T))^{-σ}; for the previously unreachable logarithmic-power shifts γ(t)=(log t)^α with α>1, it proves joint universality and a quantitative Bohr-Jessen result. If correct, the conclusion is that logarithmic-power shifts are rich enough to reproduce the full statistical behaviour of ζ, while the case α=1 (plain log t) is a genuine boundary where the method provably stops. A sympathetic reader would care because these are the slowest shifts for which value-distribution results were still open.","feed_headline":"For slow (log t)^a shifts, Riemann zeta is jointly universal","feed_subtitle":"A general theorem shows such slowly growing shifts reproduce the full value distribution, with explicit error rates.","key_machinery":"The argument rests on moment estimates for Dirichlet polynomials evaluated along a shifted curve. For a shift γ, the first derivative test converts decay of ∫ (n/m)^{iγ(t)} dt into powers of 1/γ'(T); the class F' is designed so that this decay is small and the preimage of exceptional zero-neighbourhoods is controlled. For logarithmic shifts γ(t)=(log t)^α, the derivative γ'(t)=α(log t)^{α-1}/t is small, so the paper instead uses a zero-density estimate N(σ,T)≪T^{Φ(σ)+ε} to show that the measure of t for which (log t)^α lands near a zero is o(T), as long as σ_0 > x_Φ(α_1); under RH the exceptional set vanishes unconditionally. The Fourier convergence for powers with α>1 is what separates the","core_discovery":"For 1<α_1<...<α_r, the vector (log ζ(s+i(log τ)^{α_1}),...,log ζ(s+i(log τ)^{α_r})) is jointly universal in the strip x_Φ(α_1)<σ<1, and under the Riemann hypothesis in the whole strip 1/2<σ<1. In the same range, the discrepancy between the empirical distribution of log ζ(σ+i(log t)^r) and the random Euler product limit is O(((r-1) log log T)^{-σ}). The paper also proves a unified discrepancy estimate D_{σ,γ}(T)≪(log γ(T))^{-σ} for every shift in the class F'. The border case γ(t)=log t is excluded: the key Fourier integral does not converge, and the method forces the Dirichlet polynomial length to be constant.","pith_inferences":["The method suggests a critical threshold at α=1: logarithmic shifts at or below log t likely require a different probabilistic model, perhaps with an averaging kernel that handles persistent phases, rather than the standard first-derivative test.","If the Fourier-averaging obstruction is structural, then for γ(t)=log t even the existence of a limiting distribution may fail or require Cesàro-type weights; checking the limit with such weights would be a quick test.","The zero-density threshold x_Φ(α_1) is not optimal in the unconditional statement; any future improvement of zero-density estimates directly widens the universality strip.","The joint universality result likely extends to other L-functions and to shifts of the form c (log t)^α with distinct constants, as long as linear independence of log p is preserved."],"forward_implications":["The Bohr-Jessen limit theorem holds with a quantitative rate for every shift in F', including e^t, t^a, and iterated exponentials, with the rate depending only on the size of log γ(T).","Universality under logarithmic-power shifts holds jointly for any finite number of distinct exponents α>1, so the shift can be arbitrarily slow as long as it is faster than log t.","Under RH, both universality and discrepancy estimates extend to the full strip 1/2<σ<1, matching the classical picture.","The α=1 case is shown to be a methodological barrier: the same Fourier-averaging approach cannot even produce the limit distribution for log t.","The discrepancy bound D_{σ,r}(T) ≪ ((r-1) log log T)^{-σ} degrades as r→1, quantifying exactly why r=1 is hard."],"fun_headline_variants":["Joint universality for log-power shifts of Riemann zeta","New theorem: zeta universal under log-power shifts, with rates","Logarithmic-power shifts: zeta's value distribution captured","Slow log-shifts: zeta jointly universal, with explicit error","Universality for log^a t shifts of zeta, plus error bounds"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the relevant shift has exponent α>1, so that the derivative of (log t)^α eventually dominates the reciprocal of t; if the shift is plain log t, the key average fails to converge and the proof collapses.","fun_headline_variants_meta":{"raw":{"variants":["Joint universality for log-power shifts of Riemann zeta","New theorem: zeta universal under log-power shifts, with rates","Logarithmic-power shifts: zeta's value distribution captured","Slow log-shifts: zeta jointly universal, with explicit error","Universality for log^a t shifts of zeta, plus error bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000724,"raw_usage":{"total_tokens":3017,"prompt_tokens":615,"completion_tokens":2402,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":359,"completion_tokens_details":{"reasoning_tokens":2312}},"tokens_in":359,"tokens_out":2402,"duration_ms":16560,"temperature":1.0,"reasoning_tokens":2312,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T10:58:59.534284+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find one rectangle R and one sequence T_k for which |P_{T_k}(log ζ(σ+i(log t)^r)∈R)-P(log ζ(σ,X)∈R)| grows faster than ((r-1) log log T_k)^{-σ}, with r>1 and σ>x_Φ(r); that would disprove the main quantitative theorem.","supporting_citations":[],"review_version":1}