Lower bounds for high moments of zeta sums

Hao Zhang; Weijia Wang; Zikang Dong

REVIEW 3 major objections 5 minor 15 references

Lower bounds for high moments of zeta sums

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves unconditional lower bounds for high moments of zeta sums, matching the upper bounds that were previously known only conditionally on the Riemann hypothesis.

desk verdict New unconditional lower bound for high moments of zeta sums, matching Gao's conditional upper bound, but the proof delegates the load-bearing steps to Szabó's character-sum paper and never estimates the transfer error. read the letter →

arxiv 2506.09334 v1 pith:3BQGVATK submitted 2025-06-11 math.NT

Zikang Dong , Weijia Wang , Hao Zhang This is my paper · ORCID

classification math.NT MSC 11L4011M06

keywords zetasumshighmomentsunconditionallowerboundsrandommultiplicativefunctionsDirichletpolynomialsRiemannhypothesischaracterHölder'sinequality

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

The paper proves that the high moments of the zeta sum $\sum_{n\le x} n^{-it}$, averaged over $t$ in $[0,T]$, grow at least like $x^k (\log \min\{x,T/x\})^{(k-1)^2}$ for every $k>2$, with no unproved hypothesis. This is the same order of magnitude as the previously known upper bound conditional on the Riemann hypothesis, and it matches what the random multiplicative function model predicts. The proof works by bounding a carefully chosen proxy object $R(t)$ from above in the mean and from below when weighted by the zeta sum, then applying H\"older's inequality. The lower-bound part is imported from the paper's companion treatment of character sums, after transferring the integral over $t$ to an expectation over a random multiplicative function.

What carries the argument

The proxy object is $R(t) = \sum_{|\ell|\le (\log y)/2} \prod_{m=1}^M R_{m,\ell}(t)$, where each $R_{m,\ell}(t)$ is the square of a truncated exponential in $\Re D_{m,\ell}(t)$, and $D_{m,\ell}(t)$ is a short Dirichlet polynomial over primes in the dyadic block $(y_{m-1}, y_m]$. The parameter $y = x^{1/C_0}$ splits the primes into blocks, and the truncation parameters $J_m$ are chosen so that $\prod_m y_m^{10^4 k J_m} < x$. H\"older's inequality with exponents $k$ and $k/(k-1)$ couples this proxy to the zeta sum: once the mean of $R(t)^{k/(k-1)}$ is shown to be $\ll_k (\log y)^{k^2+1}$ and the weighted mean of $|\sum_{n\le x} n^{-it}|^2 R(t)$ is shown to be $\gg_k x (\log y)^{k^2-1}$, the theorem follows.

What would settle it

Take a sequence of pairs $(T, x)$ with $x = T^{1/2}/(\log T)^{10}$ (so $L \approx x$ for large $T$) and compute the normalized moment $\frac{1}{T}\int_0^T |\sum_{n\le x} n^{-it}|^{2k}\,dt \,/\, [x^k(\log L)^{(k-1)^2}]$ for $k=3$. The theorem asserts this ratio has a positive liminf as $T\to\infty$; if the ratio tends to zero along any sequence, the central claim is false.

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Extended reading notes

Core claim

The central claim is that, for large $T$ and every $k>2$, the average $\frac{1}{T}\int_0^T |\sum_{n\le x} n^{-it}|^{2k}\,dt$ is at least a positive constant (depending only on $k$) times $x^k (\log L)^{(k-1)^2}$, where $L = \min\{x, T/x\}$. This gives the same order of magnitude as the upper bound previously obtained under the Riemann hypothesis, so the lower bound is unconditional. The exponent $(k-1)^2$ is the novel part: it comes from the product over dyadic blocks of primes in a truncated-exponential proxy, and matches the random multiplicative model's high-moment prediction. The proof reduces the theorem to two estimates on the proxy $R(t)$: an upper bound for its $(k/(k-1))$-th moment and a lower bound for its weighted average against the zeta sum squared.

Load-bearing premise

The proof assumes that the difference between the $t$-average integral and the corresponding expectation over a random multiplicative function is negligibly small, and that the character-sum lower bound the paper cites transfers to this zeta-sum setting without a loss; neither the size of that difference nor the transfer is checked in the paper.

Editorial extensions

If this is right

The high moments of zeta sums now have the same unconditional order of magnitude as the upper bound known under the Riemann hypothesis.
The lower bound holds for the full range of $x$ relative to $T$, with the logarithmic factor switching from $\log x$ to $\log(T/x)$ according to $L = \min\{x, T/x\}$.
Together with the conditional upper bound, this pins the order of magnitude of the $2k$-th moment as $x^k(\log L)^{(k-1)^2}$ for every $k>2$, assuming the Riemann hypothesis only for the upper side.
The H\"older-coupling method shows that the exponent $(k-1)^2$ arises from the product over dyadic prime blocks in the proxy, matching the random multiplicative model's prediction.

Reading between the lines

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

An explicit bound on the transfer error in Proposition 1 would make the proof self-contained and would likely yield effective implied constants in Theorem 1.1.
The same proxy-object construction should extend to shifted moments of zeta sums and to moments of derivatives of zeta functions, where currently only conditional upper bounds are known.
If this lower bound is sharp, then the high moments of zeta sums are now understood to the order of magnitude, leaving only the explicit constant open.

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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,

The short version: this paper states the expected unconditional lower bound for high moments of zeta sums, matching Gao's conditional upper bound and generalizing Szabó's character-sum lower bound. The result is very likely true. But the proof as written is not self-contained at exactly the load-bearing steps, so treat Theorem 1.1 as a plausible theorem with a sketched proof rather than a complete proof.

What is genuinely new: Theorem 1.1, the first unconditional lower bound for these moments. The strategy is standard after Szabó: Hölder with a proxy R(t) built from short Dirichlet polynomials, then a lower bound for |S|^2 R integrated, and an upper bound for R^{k/(k-1)}. Proposition 2 and Lemma 3.1 are actually proved in the text, and the argument there is clear. The definitions (subdivision ratio 20, truncation parameters J_m, the ℓ-sum) all follow Szabó's paper [14] closely, which is fine because that paper is the right template.

Where it gets soft: Proposition 1, the entire source of the lower bound, has a two-line proof. First it asserts an integral-to-expectation relation with an Error term that is never even written out, let alone bounded. For the lower bound to survive, this Error needs to be o(x (log y)^{k^2-1}). Second, after that reduction it says 'this follows immediately from the proof of Proposition 3.1 in [14]'. But the random model here is Steinhaus zeta sums, not character sums, and the parameters y = x^{1/C0}, the J_m, and the outer ℓ-sum need to be checked. The same pattern appears in Lemma 3.2, which transfers to 'Proposition 5.1 in [14]' with no details. The two preliminary lemmas (2.1, 2.2) are stated but never proved and never actually invoked in the text. So the reader who wants to verify the paper has to do the adaptation themselves.

I don't think this is a sign the result is wrong. The structure is too deliberate for that, and the missing steps are exactly the kind that Szabó's methods should supply. But it's a real gap, not a cosmetic one. A referee would need to see the Error bound and the adapted arguments before the proof is complete.

Recommendation: send it to peer review, but with an explicit request that the authors fill in the transfers from [14] and bound the error in Proposition 1. If they can do that, the paper is a solid contribution. As it stands, it's a valuable announcement with a sketch.

Referee Report

3 major / 5 minor

Summary. The paper claims an unconditional lower bound for the high moments of zeta sums: for large T and any k > 2, (1/T) ∫_0^T |∑_{n≤x} n^{-it}|^{2k} dt ≫_k x^k (log L)^{(k-1)^2}, where L = L(x,T) = min{x, T/x}. The method is to apply Hölder's inequality with a carefully chosen proxy object R(t), prove a lower bound for the weighted mean square of the zeta sum against R(t) (Proposition 1), and prove an upper bound for the integral of R(t)^{k/(k-1)} (Proposition 2). Both propositions are supposed to follow by transferring known estimates for the Steinhaus random multiplicative function from Szabó's work on character sums to zeta sums, but in the submitted manuscript the transfer is asserted rather than proved.

Significance. If the proof can be completed, Theorem 1.1 is a significant result: it would remove the Riemann hypothesis from Gao's conditional upper bound and establish the conjectured order of magnitude of high moments of zeta sums for k > 2. The high-level strategy is elegant and the paper is concise, with the proxy construction and the Hölder argument clearly laid out. However, the decisive lower-bound step is not actually proved in the manuscript: Proposition 1 rests on an unquantified error term and on an undeveloped appeal to the proof of Proposition 3.1 in [14]. The same applies to the random-model input in Lemma 3.2. The manuscript therefore currently functions as a research announcement rather than a complete proof.

major comments (3)

[Section 3, Proposition 1] The proof of Proposition 1 begins with the relation (1/T)∫_0^T |∑_{n≤x} n^{-it}|^2 R(t) dt = E |∑_{n≤x} f(n)|^2 R(f) + Error, but no estimate for Error is given. Since Proposition 1 is the only source of the lower bound in Theorem 1.1, a non-negligible or negative Error would invalidate the claimed lower bound. The author must either prove that Error = o(x (log y)^{k^2-1}) with explicit hypotheses, or replace this assertion by a fully stated and proved lemma. As written, the main theorem is unsupported at its decisive step.
[Section 3, Proposition 1] The sentence 'This follows immediately from the proof of Proposition 3.1 in [14]' is not a proof in the present manuscript. Lemmas 2.1 and 2.2, which are the stated random-model inputs, are neither proved nor invoked in the displayed argument, and the adaptation of Szabó's character-sum argument to the Steinhaus setting with the specific choices y = x^{1/C0}, the truncation parameters J_m, and the outer ℓ-sum is not demonstrated. The authors need to include the derivation or explicitly state the needed result as a theorem with all hypotheses and a proof.
[Section 3, Lemma 3.2] Lemma 3.2 is proved by first asserting an integral-to-expectation inequality and then saying that 'the lemma follows immediately from the proof of Proposition 5.1 in [14].' No error term is bounded in the passage from the zeta-sum integral to the Steinhaus expectation, and the applicability of Szabó's argument to the present R_{m,ℓ} and U_{m,ℓ} is not verified. Since Lemma 3.2 is the key input in Proposition 2, the upper bound needed for the Hölder factor is not fully justified. This is a load-bearing gap, not a presentation issue.

minor comments (5)

[Section 1] The Introduction contains typos: 'Stenhaus' should be 'Steinhaus' and 'Radmacher' should be 'Rademacher'.
[Section 2, Lemma 2.2] The displayed definition of E_{m,ℓ}(f) appears garbled: the right-hand side is written as exp(...) - R_{m,ℓ}(f) = sum over u,v, with R_{m,ℓ}(f) already used earlier as a truncated exponential. Please rewrite the definition so that the remainder term is unambiguous.
[Section 3, definition of U_{m,ℓ}] The second case in the definition of U_{m,ℓ} has a typographical formatting issue; the inequalities 'J_m/(100k) ≤ A_m ≤ 100 k J_m' and '100 k J_m ≤ A_m' should be displayed clearly so that the piecewise definition is readable.
[Section 3, displayed Hölder inequality] The displayed Hölder inequality before Proposition 1 is not numbered; numbering it would make the cross-reference in Proposition 1's proof cleaner.
[References] Reference [13] is missing journal and volume data, and the theorem quoted from Gao is labeled simply 'Theorem.' before the introduction; please number it for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lower bound is imported from external work by Harper and Szabó, not from the authors' own results; the sole self-citation is contextual.

full rationale

The paper's central derivation chain is: Hölder's inequality (3.2) reduces Theorem 1.1 to a lower bound for (1/T)∫|Σn^{-it}|^2 R(t)dt (Proposition 1) and an upper bound for (1/T)∫R(t)^{k/(k-1)}dt (Proposition 2). Neither step is circular. Proposition 1 invokes an integral-to-expectation relation and then asserts that the desired expectation lower bound 'follows immediately from the proof of Proposition 3.1 in [14]', i.e. Szabó's external character-sum paper; this is a reliance on an external result, not on the present authors' own claims. Lemma 2.1 and Lemma 2.2 are stated without proof and are not actually used in the written proof, but this is a completeness gap, not circularity. Proposition 2 uses Lemma 3.1 (proved in the paper) and Lemma 3.2, which is again taken from 'the proof of Proposition 5.1 in [14]' — again external. The only self-citation, [4] by the same authors, appears in the introduction as a general reference for zeta sums and is not load-bearing. The unestimated 'Error' term in Proposition 1 is a serious correctness concern — if Error is too large the lower bound collapses — but it is an incompleteness of proof, not a circular dependence. Under the stated rules, a missing estimate or a gap in a proof that relies on another paper's argument does not constitute circularity. No equation in the paper is shown to reduce to the target theorem by definition, no fitted parameter is renamed as a prediction, and no uniqueness claim from the authors' prior work is used to force a choice. Hence the appropriate circularity score is 0.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The paper's central theorem is built on a chain of external results: Harper's random multiplicative function moments, Gao's conditional upper bound, and Szabó's character-sum lower-bound proofs. The only new ingredient is the transfer from character sums to zeta sums, which is asserted rather than proved, so the ledger is dominated by borrowed machinery.

free parameters (3)

C0 = sufficiently large, depending only on k (not numerically specified)
Used to set y = x^{1/C0} and J_M = C0/(10^5 k); chosen by hand so that inequality (3.1) and J_M ≥ exp(10^4 k^2) hold.
Subdivision ratio 20 = 20
The dyadic-style subdivision y_{m-1}=y_m^{1/20} is chosen by hand; the proof's estimates depend on this geometric progression.
Truncation parameters J_m = J_1=(log log y)^{3/2}, J_M=C0/(10^5 k), J_m=J_M+M-m
Introduced ad hoc to define the proxy R(t) and to satisfy Lemma 3.1 and the product inequality (3.1).

assumptions (3)

ad hoc to paper The Steinhaus random multiplicative function expectation accurately represents the zeta-sum mean square in Proposition 1, with negligible error term.
Asserted in Section 3 without proof; the size of Error is not quantified and the lower bound is imported from [14].
domain assumption The lower-bound machinery in Szabó [14] (Proposition 3.1 and Proposition 5.1) transfers unchanged to zeta sums.
Lemma 3.2 and Proposition 1 are justified by 'follows immediately' from [14]; this is a substantive assumption about the zeta-sum setting.
standard math Harper's high-moment estimates for Steinhaus random multiplicative functions, quoted in Lemmas 2.1 and 2.2, are correct.
External results from [8] and [14] are used without proof; they are treated as black boxes.

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Cite this review

Pith. "Pith review of Lower bounds for high moments of zeta sums." pith.science (2026). https://pith.science/paper/3BQGVATK

@misc{pith2026250609334,
  author       = {Pith},
  title        = {Pith review of: Lower bounds for high moments of zeta sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3BQGVATK}},
  note         = {Machine review of arXiv:2506.09334}
}

abstract

In this article, we investigate high moments of zeta sums $\sum_{n\le x}n^{i t}$. We show unconditional lower bounds for them.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 13 canonical work pages

[14]

Szab´ o,A lower bound on high moments of character sums, arXiv:2409.13436, 2024

B. Szab´ o,A lower bound on high moments of character sums, arXiv:2409.13436, 2024

arXiv 2024
[1]

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Cochrane and Z. Zheng,High order moments of character sums, Proc. Amer. Math. Soc.126 (1998), no. 4, 951–956

work page 1998
[2]

M. J. Curran,Correlations of the Riemann zeta function, arXiv:2303.10123, 2023

work page Pith review arXiv 2023
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de la Bret` eche, M

R. de la Bret` eche, M. Munsch and G. Tenenbaum,Small G´ al sums and applications, J. Lond. Math. Soc.103(2021), no. 1, 336–352

work page 2021
[4]

Z. Dong, W. Wang and H. Zhang,Large zeta sums, to appear in Period. Math. Hung

work page
[5]

Gao,Upper bounds for moments of zeta sums, J

P. Gao,Upper bounds for moments of zeta sums, J. number Theory,278(2026), 47–63

work page 2026
[6]

Granville and K

A. Granville and K. Soundararajan,Large character sums, J. Amer. Math. Soc.14(2001), 365– 397

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A. J. Harper,Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos, Forum Math. Pi, 2020;8:e1

work page 2020

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[8]

A. J. Harper,Moments of random multiplicative functions, II: high moments, Algebra Number Theory13(2020), no. 10, 2277–2321

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[9]

A. J. Harper,The typical size of character and zeta sums iso( √x), ArXiv:2301.04390

arXiv
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Munsch,Shifted moments of L-functions and moments of theta functions, Mathematika63 (2017), no

M. Munsch,Shifted moments of L-functions and moments of theta functions, Mathematika63 (2017), no. 1, 196–212

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M. Munsch and I. E. Shparlinski,Upper and lower bounds for higher moments of theta functions, Q. J. Math.67(2016), no. 1, 53–73

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2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [14]

Szab´ o,A lower bound on high moments of character sums, arXiv:2409.13436, 2024

B. Szab´ o,A lower bound on high moments of character sums, arXiv:2409.13436, 2024

arXiv 2024

[2] [1]

Zheng,High order moments of character sums, Proc

Cochrane and Z. Zheng,High order moments of character sums, Proc. Amer. Math. Soc.126 (1998), no. 4, 951–956

work page 1998

[3] [2]

M. J. Curran,Correlations of the Riemann zeta function, arXiv:2303.10123, 2023

work page Pith review arXiv 2023

[4] [3]

de la Bret` eche, M

R. de la Bret` eche, M. Munsch and G. Tenenbaum,Small G´ al sums and applications, J. Lond. Math. Soc.103(2021), no. 1, 336–352

work page 2021

[5] [4]

Z. Dong, W. Wang and H. Zhang,Large zeta sums, to appear in Period. Math. Hung

work page

[6] [5]

Gao,Upper bounds for moments of zeta sums, J

P. Gao,Upper bounds for moments of zeta sums, J. number Theory,278(2026), 47–63

work page 2026

[7] [6]

Granville and K

A. Granville and K. Soundararajan,Large character sums, J. Amer. Math. Soc.14(2001), 365– 397

work page 2001

[8] [7]

A. J. Harper,Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos, Forum Math. Pi, 2020;8:e1

work page 2020