Distribution of sums involving Dirichlet characters over the $k$-free integers

Caio Bueno

REVIEW 3 major objections 4 minor 22 references

Distribution of sums involving Dirichlet characters over the $k$-free integers

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read 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

desk verdict 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. read the letter →

arxiv 2602.23100 v2 pith:5MLOL3IJ submitted 2026-02-26 math.NT

Caio Bueno This is my paper · ORCID

classification math.NT MSC 11M0611M2611N3711K65

keywords Dirichletcharactersk-freeintegerslimitingdistributionnegativediscretemomentszetafunctionderivativemodifiedlargedeviationsRiemannhypothesis

topics The Riemann Hypothesis

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 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.

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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)}.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

Short 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.

What 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.

The 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.

Soft spots. First and largest: Items (i)/(ii) of Theorem 1.1 assume Σ_{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.

Second, 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.

Section 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.

Bottom 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.

Best,

Referee Report

3 major / 4 minor

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.

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 (3)

[Lemma 4.4] 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.
[Lemma 4.3, vertical contour bound] 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.
[Theorem 1.2, proof after Eq. (8)] 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.

minor comments (4)

[Lemma 4.1 and Theorem 1.1] 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.
[Lemma 4.1 proof] 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 2, historical discussion] 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.
[Theorem 6.1 and Conjecture 6.3] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: Theorems are conditional on explicitly stated negative-moment hypotheses; no target quantity is used as an input, and self-citations are not load-bearing.

full rationale

The paper's main theorems are conditional statements: given GRH plus Items i/ii (negative discrete moment bounds), they prove a limiting distribution and an exceptional-set bound. The moment bounds enter Lemma 4.1 only to verify Assumption 1 of Ng's Proposition 3.1 (the required theta<2), and then appear in Lemma 4.4 and Theorem 1.2 through Cauchy-Schwarz. This is a standard transfer from an assumed zero-moment estimate to a partial-sum estimate; the target bound x^{1/(2k)}(log x)^{1/2+epsilon} is not used to set the exponent in Items i/ii, and the proof never fits a parameter to the conclusion. Section 8 honestly states that no upper bounds for the negative moments are currently established; this makes the theorems conditional on an unproved conjecture, which is a correctness/risk issue rather than circularity. The self-citations ([3], [4]) appear in motivational remarks and in the statement of Conjecture 6.3, which strengthens the authors' earlier conjecture; they are not used as load-bearing support for the derivations. The large-deviation section explicitly says it is 'identical to' Meng's k-free number bounds and reuses Ng/Meng's framework, an external, acknowledged source. I therefore find no specific circular step.

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

No fitted constants; all assumptions are named conjectures or standard hypotheses. The modified character gχ and the finite product P(s) are not newly postulated physical entities; they are defined in the paper but already studied in prior work by the author.

assumptions (4)

domain assumption Generalized Riemann hypothesis (GRH) for ζ and Dirichlet L-functions
Assumed throughout (Theorem 1.1, Lemmas 4.1–4.4) to control horizontal/vertical contour integrals and zero-density; unproved.
ad hoc to paper Discrete negative second-moment bounds: Σ_{0<γ≤T}|ζ'(ρ)|^{-2} ≪ T^{1+1/k-ε} (and the L-analogue)
Items i/ii of Theorem 1.1; used in Lemma 4.1 and Lemma 4.4 to make the tail of the zero sum negligible. Section 8 states no upper bounds for these moments are currently known; a stronger positivity (simplicity of zeros) is implied.
domain assumption Linear Independence (LI/GLI) of imaginary parts of zeros
Section 6 only, for the Fourier transform and large deviations; unproved and very strong.
domain assumption Simplicity of the non-trivial zeros of ζ(ks) and L(ks,χ)
Explicitly assumed in Lemma 4.3; follows from the assumed finiteness of the negative second moment, but stated separately.

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Cite this review

Pith. "Pith review of Distribution of sums involving Dirichlet characters over the $k$-free integers." pith.science (2026). https://pith.science/paper/5MLOL3IJ

@misc{pith2026260223100,
  author       = {Pith},
  title        = {Pith review of: Distribution of sums involving Dirichlet characters over the $k$-free integers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MLOL3IJ}},
  note         = {Machine review of arXiv:2602.23100}
}

abstract

Assuming the generalized Riemann hypothesis and a bound for the negative discrete moments of the Riemann zeta function (resp. Dirichlet $L$-functions), we prove the existence of a logarithmic limiting distribution for the normalized partial sums $x^{-\frac{1}{2k}}\sum_{n\leq x}f(n)$, where $f$ is either a quadratic Dirichlet character or a modified Dirichlet character, restricted to the $k$-free integers. Moreover, we strengthen a conjecture made by Aymone, Medeiros and the author (cf. Ramanujan J. 59(3):713-728, 2022) concerning the precise order of magnitude for these partial sums.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

Limiting distributions of the classical error terms of prime number theory

A. Akbary, N. Ng, and M. Shahabi. “Limiting distributions of the classical error terms of prime number theory”. In:Q. J. Math.65.3 (2014), pp. 743–780.issn: 0033-5606,1464-3847.doi:10.1093/qmath/hat059.url:https://doi.org/10. 1093/qmath/hat059

work page doi:10.1093/qmath/hat059.url:https://doi.org/10 2014

[2] [2]

A note on multiplicative functions resembling the M¨ obius function

M. Aymone. “A note on multiplicative functions resembling the M¨ obius function”. In:J. Number Theory212 (2020), pp. 113–121.issn: 0022-314X,1096-1658.doi:10. 1016/j.jnt.2019.10.025.url:https://doi.org/10.1016/j.jnt.2019.10.025

work page doi:10.1016/j.jnt.2019.10.025 2020

[3] [3]

Multiplicative functions supported on thek-free integers with small partial sums

M. Aymone, C. Bueno, and K. Medeiros. “Multiplicative functions supported on thek-free integers with small partial sums”. In:Ramanujan J.59.3 (2022), pp. 713– 728.issn: 1382-4090,1572-9303.doi:10.1007/s11139-022-00618-9.url:https: //doi.org/10.1007/s11139-022-00618-9

work page doi:10.1007/s11139-022-00618-9.url:https: 2022

[4] [4]

Modified Dirichlet Character Sums over thek-free Integers

C. Bueno. “Modified Dirichlet Character Sums over thek-free Integers”. In:Bull. Braz. Math. Soc. (N.S.)56.2 (2025), Paper No. 28.issn: 1678-7544,1678-7714.doi: 10.1007/s00574-025-00453-8.url:https://doi.org/10.1007/s00574-025- 00453-8

work page doi:10.1007/s00574-025-00453-8.url:https://doi.org/10.1007/s00574-025- 2025

[5] [5]

Lower bounds for negative moments ofζ ′(ρ)

P. Gao and L. Zhao. “Lower bounds for negative moments ofζ ′(ρ)”. In:Mathematika 69.4 (2023), pp. 1081–1103.issn: 0025-5793,2041-7942.doi:10.1112/mtk.12219. url:https://doi.org/10.1112/mtk.12219. 27

work page doi:10.1112/mtk.12219 2023

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