{"id":"a7896eba-2044-48e7-8995-58ace20b8eb4","arxiv_id":"2602.23100","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Conditional on GRH and negative moment bounds, e^{-y/(2k)}∑_{n≤e^y} μ^(k)(n)χ(n) (or a modified character) has a logarithmic limiting distribution, and the sums obey an almost-all O(x^{1/(2k)}(log x)^{1/2+ε}) bound.","lead":"Under the generalized Riemann hypothesis and a strong unproved bound on negative moments of zeta and L-functions, this paper proves that normalized partial sums of quadratic characters over k-free integers have a limiting distribution, and that the sums are almost always as small as x^{1/(2k)} times a log factor. This extends a line of work on limiting distributions for Möbius and k-free summatory functions and sharpens a 2022 conjecture by the author and collaborators.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorems 1.1/1.2 rest on unproved negative-moment bounds (Items i/ii); these are load-bearing because Lemma 4.1 fails if the exponent is even slightly weakened, and Section 8 concedes no upper bounds are known.","rationale":"I read the paper in good faith as a serious conditional result in the framework of Ng, Meng, Humphries, and Akbary–Ng–Shahabi. I checked the contour calculations in Lemma 4.3, the use of Proposition 3.1 in Lemma 4.1, the error terms in Lemma 4.4, and the application of Theorem 3.2 in Theorem 1.1. I did not find a clear internal inconsistency that would invalidate the proof. In particular, the apparent awkwardness in Theorem 1.1 — where the tail T ≤ |γ| < e^Y is bounded separately — can be resolved by taking the truncation parameter X in Theorem 3.2 to be e^Y, absorbing the tail into the main sum, and bounding only the divided error term ̃E(e^y,e^Y)/e^{y/(2k)}, whose mean-square does tend to 0. The one ineliminable weakness is the negative-moment hypothesis, which the paper itself flags in Section 8 as having no currently established upper bounds. Since the theorems are explicitly conditional and the hypothesis is precisely stated, the correct disposition is conditional acceptance, exactly as the reader concluded. No verdict change is needed.","tokens_in":18982,"tokens_out":35904,"duration_ms":307901,"concrete_test":"Set J_{-1}(T) ≪ T^α and recompute the exponent in the proof of Lemma 4.1. Assumption 1 holds only when α + 1 − 1/k < 2, i.e. α < 1 + 1/k. Taking the limiting case α = 1 + 1/k + ε, the bound from Proposition 3.1 becomes O(1) instead of O(1/T^ε), so the mean-square of the tail in Theorem 1.1 fails to tend to 0. This analytically verifies that the theorem's validity is tied precisely to the assumed −ε in Items i/ii; if a future result supplied only α = 1 + 1/k + ε, the argument would collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single load-bearing point is Items i/ii of Theorem 1.1: the discrete negative-moment bounds Σ_{0<γ≤T}|ζ'(ρ)|^{-2} ≪ T^{1+1/k−ε} and their L-analogues. Section 8 states plainly that no upper bounds for these moments are currently established, so the theorems are conditional on a specialized conjecture, not on GRH alone. These bounds enter twice: in Lemma 4.1 they force the zero-tail exponential sums to have mean-square O(1/T^ε) via Proposition 3.1, and in Lemma 4.4 they control the error when passing from good heights T_n to arbitrary T. If the true exponent were 1+1/k+ε instead of 1+1/k−ε, Assumption 1 would fail: with λ_n=γ_n/k and r_n=L(ρ_n/k,χ)/(ρ_n L'(ρ_n,χ^k)), the sum Σ λ_n^2|r_n|^2 would be ≍ T^{2+ε'+ε}, giving θ>2, so Proposition 3.1 would yield only O(1) and the tail mean-square in Theorem 1.1 would not vanish. Thus the central claim is exactly as conditional as the stated moment bound. This is not an internal contradiction, but it is the least secure point in the proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies partial sums of μ^(k)χ and μ^(k)gχ, where χ is a primitive quadratic character and gχ is the modified character that sets χ(p)=1 for p|q. Under GRH and additional unproved negative discrete moment bounds for ζ' or L' (Items i/ii of Theorem 1.1), the author proves that e^{-y/(2k)} ∑_{n≤e^y} f(n) has a logarithmic limiting distribution, and that ∑_{n≤x} f(n) ≪_ε x^{1/(2k)} (log x)^{1/2+ε} except on a set of finite logarithmic measure. The paper also gives large-deviation estimates under LI/GLI and a weak-Mertens-type mean-square result. The proofs follow the framework of Ng and Meng, with an explicit formula obtained by contour integration in the half-plane Re(s)>0.","tokens_in":19398,"tokens_out":31346,"duration_ms":257262,"significance":"If the results are correct, they provide the first limiting-distribution characterization for character sums over k-free integers at the conjectured square-root-type scale, and they sharpen a conjecture of Aymone–Medeiros–the author. The adaptation of Ng's B_2-almost-periodicity machinery to this setting, together with the detailed treatment of the modified character gχ, is a useful contribution. The main caveat is that the central theorems are conditional not only on GRH but also on the specialized negative-moment bounds in Items i/ii, which Section 8 explicitly says are not yet established. The paper is honest about this, but it means the headline results are exactly as conditional as those moment bounds.","major_comments":[{"comment":"The last term in the stated error term is x^{1/(2k)}(log T)^{1/2} T^ε. In the proof, however, the final line gives x^{1/(2k)}(log T)^{1/2}/T^ε (one obtains (log T)^{1/2} T^{-ε} after the Cauchy–Schwarz step). With the printed sign, the mean-square integral in Theorem 1.1 would contain a term Y e^{2Yε} that does not vanish after division by Y. The later use in Theorem 1.2, where the error is written as x^{1/(2k)-ε}(log x)^{1/2} for T≍x, is consistent only with the T^{-ε} version. This sign error is load-bearing and must be corrected.","section":"Lemma 4.4"},{"comment":"In the proof of Lemma 4.3, after setting σ1=ε, the vertical-side integral is bounded by x^{σ1} T^{σ1(k−1)+2ε} = x^ε T^{ε(k+1)}. The displayed final error term x^ε T^ε is therefore missing a factor T^{ε(k−1)}. This error propagates to Lemma 4.4 and to Theorem 1.2. The main theorems can still be made to work by choosing ε sufficiently small (e.g. ε < 1/(2k(k+2)) in the mean-square integral), but the statements as written are false and the proofs need a corrected record of the vertical contribution.","section":"Lemma 4.3, vertical contour bound"},{"comment":"The estimate for the 'smaller sum' over |γ|<log T is not justified as written. The displayed bound is x^{1/(2k)}(log T)^{1/2−1/(2k)}(log T) ∑_{0<γ<log T} |ζ'(ρ)|^{-1}. Inserting the bound ∑ |ζ'(ρ)|^{-1} ≪ (log T)^{1+1/(2k)−ε/2}(log log T)^{1/2} gives (log T)^{5/2+o(1)}, not (log T)^{1/2+ε}. A correct treatment can be obtained by partial summation with S(t)=∑_{γ≤t}|ζ'(ρ)|^{-1} and b(t)=t^{−1/2−1/(2k)}, which yields (log T)^{1/2−ε/2+o(1)}; the theorem is salvageable, but the proof as written has a gap at this load-bearing point.","section":"Theorem 1.2, proof after Eq. (8)"}],"minor_comments":[{"comment":"Lemma 4.1 is stated for L(s,χ^k), while Theorem 1.1's Item i for even k concerns ζ'(ρ). Since χ is quadratic, χ^k is principal for even k and the relevant zeros are those of ζ(s). This reduction should be stated explicitly in Lemma 4.1 or in the application.","section":"Lemma 4.1 and Theorem 1.1"},{"comment":"The bound |L(ρ/k,χ)|/|ρ| ≪ |γ|^{−1/2−1/(2k)+ε} is used without comment. It follows from the functional equation (3), the gamma-factor bound (4), and the Lindelöf bound for L(1−ρ/k,χ). Stating this would improve readability.","section":"Lemma 4.1 proof"},{"comment":"The paper credits Ng [21] and Meng [14] appropriately, but the sentence introducing Akbary–Ng–Shahabi's B_p-almost-periodicity criterion says it was 'known since 1930s'; a reference or more precise attribution would be helpful.","section":"Section 2, historical discussion"},{"comment":"The large-deviation theorem changes the moment hypothesis from Items i/ii to ∑|ζ'(ρ)|^{-2}≪T^{1+ε}. This distinction is clear but should be explicitly noted, since the main theorems use the stronger T^{1+1/k−ε} bound.","section":"Theorem 6.1 and Conjecture 6.3"}],"recommendation":"major_revision","confidential_remarks":"The paper's main theorems are conditional on a specialized, currently unproved negative-moment conjecture, and Section 8 is frank about this. The technical errors in Lemmas 4.3–4.4 and Theorem 1.2 are repairable, and the overall framework is sound. I would recommend major revision rather than rejection; the authors should correct the displayed error-term exponents and rewrite the small-sum estimate in Theorem 1.2."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nShort version: a genuinely new, competently executed conditional result, but the theorems stand on a negative-discrete-moment bound that is currently not known at all, and there is one technical step in the proof of Theorem 1.1 I could not verify as written.\n\nWhat is new: the first limiting distribution, in the logarithmic sense, for normalized partial sums of quadratic Dirichlet characters and modified characters restricted to the k-free integers, plus an almost-all bound at x^{1/(2k)} (log x)^{1/2+ε}, sharpening the Aymone–Bueno–Medeiros conjecture. The framework is Ng/Meng/Humphries almost-periodicity, and the main technical novelty is handling the factor P(s)=∏_{p|q}(1-p^{-s})^{-1}, whose poles force the contour to stay in Re(s)>0. That is a real complication and the author handles it sensibly.\n\nThe proof is mostly clear and careful. The explicit formula (Lemmas 4.3 and 4.4) and the verification of Assumptions 1–3 for the zero sequences are standard but needed doing in this context. The paper is also honest about what is conditional.\n\nSoft spots. First and largest: Items (i)/(ii) of Theorem 1.1 assume\nΣ_{0<γ≤T}|ζ'(ρ)|^{-2} ≪ T^{1+1/k-ε} (and the L-analogue). As Section 8 admits, no upper bound of this type is known; the only unconditional bound under GRH and simplicity is T^{2+ε}, which is far too weak. The stress-test is right that this is load-bearing: in Lemma 4.1 it enters exactly to make the exponent θ in Assumption 1 less than 2. If the true exponent were 1+1/k+ε, the tail mean-square would not vanish and Theorem 1.1 would collapse. So this is a conditional result on a specialized conjecture, not just on GRH. The author says so, but readers should not mistake it for a GRH-conditional theorem.\n\nSecond, the proof of Theorem 1.1 integrates an error term after dividing by e^{y/(2k)} and claims the result is O(1). I did not see how the x^{1/(2k)}(log T)^{1/2}T^ε term from Lemma 4.4 is handled uniformly as Y grows; the choice of T relative to e^Y is not spelled out. Similarly, the horizontal contour estimate in the proof of Lemma 4.3 has a displayed exponent that looks off: the term x^{1/2}/T^{1/2-2ε+σ1} does not obviously follow from the preceding integrals. This is probably repairable, but it needs to be written out.\n\nSection 6 is a sketch, and it is labeled as such. The large-deviation results are consistent with Meng's, and the conjectural tail in Conjecture 6.3 is a reasonable extrapolation.\n\nBottom line: for anyone working on character sums or summatory functions of multiplicative functions, this is a useful contribution and deserves a serious referee. I would not desk-reject it, but I would ask for a careful check of the moment-bound dependence and the mean-square integration before accepting it.\n\nBest,","headline":"Solid conditional extension of Ng–Meng machinery to character sums over k-free integers; load-bearing negative-moment assumptions are unproved and at least one contour estimate needs a closer look.","tokens_in":19753,"tokens_out":2844,"would_cite":true,"duration_ms":28368,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26","11N37","11K65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, assuming the generalized Riemann hypothesis and a specialized unproved bound on negative moments of zeta derivatives, the normalized partial sums of Dirichlet characters over k-free integers have a limiting distribut","keywords":["Dirichlet characters","k-free integers","limiting distribution","negative discrete moments","zeta function derivative","modified Dirichlet characters","large deviations","Riemann hypothesis"],"falsifier":"A proof or computation producing infinitely many T with ∑_{0<γ≤T}|ζ'(ρ)|^{-2} ≥ T^{1+1/k} would invalidate the paper's key input. Alternatively, for a fixed k and small modulus, numerically computing e^{-y/(2k)}∑_{n≤e^y} f(n) over a long y-range and finding that its logarithmic average does not settle to a stable distribution would contradict Theorem 1.1.","tokens_in":18919,"feed_emoji":"📊","tokens_out":7053,"duration_ms":61427,"temperature":0.7,"pith_summary":"This paper aims to show that the partial sums of two arithmetic functions — a real Dirichlet character and a modified version of it, restricted to the k-free integers — are statistically well behaved once normalized by x^{1/(2k)}. Concretely, assuming the generalized Riemann hypothesis plus a particular unproved bound on the sum of reciprocal derivatives of zeta (or the associated Dirichlet L-function) over its zeros, the normalized sums e^{-y/(2k)}∑_{n≤e^y} f(n) have a limiting distribution in the sense of logarithmic averages. The same assumptions yield an almost-everywhere bound |∑_{n≤x} f(n)| ≪ x^{1/(2k)} (log x)^{1/2+ε}, except on a set of finite logarithmic measure, strengthening a conjecture that the true order of magnitude is the same as the conjectured error term in summatory functions of k-free integers. A sympathetic reader would care because the result converts the erratic oscillation of these character sums into a probability law and pins down the fluctuations at the conjectured square-root-type scale, the same scale that appears for the Möbius function and k-free numbers.","feed_headline":"Character sums over k-free integers get a limiting law","feed_subtitle":"Under GRH and an unproved zero-moment bound, these character sums almost always sit at the conjectured x^{1/(2k)} scale.","key_machinery":"The key object is the explicit zero expansion ∑_{|γ|<T} L(ρ/k, χ)/(ρ Z_f(ρ)) x^{ρ/k} for the partial sums, where ρ=1/2+iγ runs over non-trivial zeros and Z_f is a rational factor (e.g., P(s)/ζ'(s) for even k, 1/L'(s,χ) for odd k, and variants for modified characters). The load-bearing input is the negative second-moment bound ∑|ζ'(ρ)|^{-2} ≪ T^{1+1/k-ε} (or the L-analogue), used in Lemma 4.1 to make the zero-tail square-integrable; from there, a general almost-periodicity criterion yields the limiting distribution, and a Cauchy-Schwarz argument controls the error when passing from good heights to arbitrary T. The finite product P(s) over primes dividing q is analytic in Re(s)>0 but has poles","core_discovery":"The central claim is Theorem 1.1: under the generalized Riemann hypothesis and the negative-moment bound ∑_{0<γ≤T}|ζ'(ρ)|^{-2} ≪ T^{1+1/k-ε} (or the corresponding L-function version for odd k), the function y ↦ e^{-y/(2k)}∑_{n≤e^y} f(n) has a limiting distribution ν_k on R. Theorem 1.2 strengthens the earlier conjecture: except on a set of finite logarithmic measure, ∑_{n≤x} f(n) ≪_ε x^{1/(2k)} (log x)^{1/2+ε}. The same machinery, with additional linear-independence assumptions, gives a large-deviation estimate for ν_k([V,∞)) and a conjectured precise oscillation with iterated logarithms. The argument's engine is an explicit formula expressing the partial sum as a sum over zeros of the match","pith_inferences":["Inference: the same method likely applies to other ratios of L-functions, e.g., sums of a non-real Dirichlet character over k-free integers, with the normalization still x^{1/(2k)} and the only new requirement being a corresponding negative-moment bound for the relevant L-function derivative.","Inference: the paper's choice to keep the contour in Re(s)>0, forced by the poles of P(s), suggests a structural limitation: for moduli with several distinct prime factors, any argument shifting the contour left of the imaginary axis would have to handle large pole contributions, so removing the negative-moment assumption may require a genuinely different mechanism.","Inference: the large-deviation predictions are coarse enough that a numerical experiment with a fixed small k and modulus could test the exponential rate exp(-c V^{2k/(k-1)}); a clear mismatch would point to either the negative-moment bound or the linear-independence assumption as the false premise."],"forward_implications":["If Theorem 1.1 is correct, the normalized sums have a well-defined logarithmic distribution, so statements like 'the sum exceeds V x^{1/(2k)} for a positive fraction of log-scales' become meaningful for every V.","Theorem 1.2 shows the partial sums cannot be much larger than x^{1/(2k)}: the exceptional set where they exceed x^{1/(2k)} (log x)^{1/2+ε} has finite logarithmic measure, so on almost all scales the sum is at the conjectured order.","Under the linear-independence conjectures, the Fourier transform of ν_k is an explicit product of Bessel functions over zeros, and the large-deviation bounds give quantitative tail estimates supporting a sharp iterated-logarithm conjecture for the maximal oscillation.","The weak-Mertens-type result ∫ (∑_{n≤x} f(n)/x^{1/(2k)})^2 dx/x ≪ log X, together with the asymptotic ∼β_k log X, gives an averaged second-moment statement consistent with fluctuations of size x^{1/(2k)}."],"fun_headline_variants":["Limiting law for k-free character sums under GRH","GRH and a moment bound give limit law for k-free character sums","Stronger conjecture: k-free character sums now have a limiting distribution","Under GRH, k-free character sums get a proven limiting law","Logarithmic limit law for Dirichlet character sums on k-free numbers"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The theorems stand on the unproved upper bound ∑_{0<γ≤T}|ζ'(ρ)|^{-2} ≪ T^{1+1/k-ε} (and its L-function analogue); without it, the central square-integrability and error-control lemmas fail.","fun_headline_variants_meta":{"raw":{"variants":["Limiting law for k-free character sums under GRH","GRH and a moment bound give limit law for k-free character sums","Stronger conjecture: k-free character sums now have a limiting distribution","Under GRH, k-free character sums get a proven limiting law","Logarithmic limit law for Dirichlet character sums on k-free numbers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001297,"raw_usage":{"total_tokens":5108,"prompt_tokens":700,"completion_tokens":4408,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":444,"completion_tokens_details":{"reasoning_tokens":4317}},"tokens_in":444,"tokens_out":4408,"duration_ms":30240,"temperature":1.0,"reasoning_tokens":4317,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T20:30:14.464140+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A proof or computation producing infinitely many T with ∑_{0<γ≤T}|ζ'(ρ)|^{-2} ≥ T^{1+1/k} would invalidate the paper's key input. Alternatively, for a fixed k and small modulus, numerically computing e^{-y/(2k)}∑_{n≤e^y} f(n) over a long y-range and finding that its logarithmic average does not settle to a stable distribution would contradict Theorem 1.1.","supporting_citations":[],"review_version":1}