Large values of Dirichlet polynomials with multiplicative coefficients

Hao Zhang; Shengbo Zhao; Weijia Wang; Yutong Song; Zikang Dong

REVIEW 2 major objections 3 minor 14 references

Large values of Dirichlet polynomials with multiplicative coefficients

T0 review · 2 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read For multiplicative-coefficient Dirichlet polynomials, this paper proves large-value lower bounds across nearly the whole range exp((logT)^(1/2+ε)) ≤ N ≤ √T.

desk verdict Genuinely new zeta-sum bound, but Theorem 1.2's multiplicative-function class is a phantom: F(c) contains only f≡1. read the letter →

arxiv 2509.09771 v1 pith:SURD76Q4 submitted 2025-09-11 math.NT

Zikang Dong , Yutong Song , Weijia Wang , Hao Zhang , Shengbo Zhao This is my paper · ORCID

classification math.NT MSC 11L4011N25

keywords DirichletpolynomialscompletelymultiplicativefunctionsresonancemethodGCDsumsOmegaresultslargevalueszetamean

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 new Omega-type lower bounds for the maximum over 1 ≤ t ≤ T of Dirichlet polynomials ∑_{n≤N} f(n)n^{it}, where f is completely multiplicative and |f(n)|=1. In the main range, where N lies between exp((logT)^(1/2+ε)) and √T, and the coefficients satisfy a positive-correlation condition Re(f(n) overline{f(m)}) ≥ c for all m,n, the maximum is at least √N exp((√2+o(1))√(log(T/N) log_3(T/N)/log_2(T/N))). In the transition range logN = √(logT log_2 T) τ with τ=(log_2T)^{O(1)}, the bound √N exp((1+o(1))A(τ+τ')√(logT/log_2T)) holds for every completely multiplicative f with |f|=1. The proofs combine a resonance construction with estimates for GCD sums, and the key technical task is showing that off-diagonal terms in the second moment are negligible.

What carries the argument

The resonator. For Theorem 1.2 it is constructed by taking a set M of ⌊T/N⌋ integers that maximizes the GCD sum (1/|M|)∑_{m,n∈M} √((m,n)/[m,n]), then thinning it to a well-separated subset M' by keeping the smallest element of each dyadic logarithmic bin and assigning weight r(m_j)=|M_j|^{1/2}. The Gaussian weight Φ(t)=e^{-t²/2} and its positive Fourier transform make the second moment a sum of terms Φ(T/logT · log(ma/nb)), whose rapid decay forces diagonal terms (am=bn) to dominate. For Theorem 1.1 the resonator uses a completely multiplicative r supported on primes in [λ², exp((logλ)²)], following Hough's construction, and the Gaussian factor bΦ(cN/logT) makes the off-diagonal contribution

What would settle it

Compute, for the r defined in Theorem 1.1 and logN = √(logT log_2 T), the off-diagonal sum ∑_{k≤N}∑_{m,n≤x, km≠n} f(km) overline{f(n)} r(m)r(n) Φ(T/logT log(km/n)) and compare it with ∑_{m≤x} r(m)^2. If this ratio does not tend to 0 as T→∞, the proof of Theorem 1.1 fails; a numerical check for moderate T would already indicate whether the claimed negligibility holds.

Watch

Extended reading notes

Core claim

The central claim is that the resonance method can be made to work across the entire range N ≤ √T for Dirichlet polynomials with multiplicative coefficients, not just in the previously treated regime where N is a small power of T. For the class F(c), the paper establishes the lower bound with the precise shape √N exp((√2+o(1))√(log(T/N) log_3(T/N)/log_2(T/N))), uniformly in N between exp((logT)^(1/2+δ)) and √T. In the transition regime where logN = √(logT log_2 T) τ, it shows the bound holds for all completely multiplicative f with |f(n)|=1, with the constant A(τ+τ') defined through the exponential-integral relations τ=∫_A^∞ e^{-u}/u du and τ'=∫_A^∞ e^{-u}/u² du. The constant √2 appears beca

Load-bearing premise

For Theorem 1.1, the load-bearing step is the claim that the off-diagonal contribution to the second moment I2(R,T) is negligible; the text justifies this as 'similar to' the I1 estimate, but the phase-gap argument there only yields a factor 1/logT, and the claim is actually saved by an unstated property: the resonator's support contains only odd integers, so their neighbours at distance one have r=0.

Editorial extensions

If this is right

If correct, these bounds show that the maximum of |∑_{n≤N} f(n)n^{it}| is at least √N times an exponential factor growing with log(T/N) throughout the near-complete range exp((logT)^(1/2+ε)) ≤ N ≤ √T.
The range of N covered improves on earlier resonance-method results, which required N ≍ T^{C(N)} with C(N) growing like a small power of logN.
For f≡1, Theorem 1.1 gives a lower bound for the classical zeta sum max_{t≤T}|∑_{n≤N}n^{it}| in the transition regime that is stronger than the previous best.
The appearance of the constant √2, the same constant as in maximal GCD sums, indicates that extreme values of these Dirichlet polynomials and extreme GCD sums are governed by the same arithmetic mechanism.
The bound in Theorem 1.2 is uniform over the class F(c), meaning a single resonator works for all such coefficients f.

Reading between the lines

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

The positive-correlation condition in F(c) is used only to keep off-diagonal terms nonnegative; a natural test is whether the same bound survives without it, by handling sign changes with a different weighting.
The proof of Theorem 1.1 depends on an unstated support property of the resonator r — all primes in its support exceed λ², so all supported integers are odd and their distance-one neighbours carry weight zero. Verifying this property explicitly is a concrete step that would close the gap left by the text's analogy to the I1 estimate.
The same resonance-plus-GCD-sum scheme may extend to other objects, such as character sums or short Dirichlet polynomials, wherever a diagonal term can be isolated by rapid decay of a weight.

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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

First, the punchline. The paper has a genuinely new lower bound for the classical zeta sum in the range exp((logT)^{1/2+ε}) ≤ N ≤ √T, and a transition-regime Omega result for arbitrary completely multiplicative f. But the advertised generality of Theorem 1.2 collapses: the class F(c) defined by Re f(n)f(m) ≥ c contains no nonconstant function. With m=1, n=p^k, the condition forces Re f(p)^k ≥ c for every k, which cannot hold for f(p)=e^{iθ} with θ≠0. Hence F(c) is either empty (c>1) or the singleton {1}. So 'for all f∈F(c)' is really 'for f=1'. That's not a mild 'additional request'—it defeats the stated improvement over Xu–Yang's uniform result.

What's good: Theorem 1.1 is a reasonable new result, and the f=1 case of Theorem 1.2 goes beyond [14], whose parameter range excludes N near √T. The machinery—Hough's sums, the de la Bretèche–Tenenbaum GCD lemma, the resonance method—is appropriate, and the arithmetic is not circular. The self-citation is a baseline, not a crutch.

Soft spots in proportion: the 'similarly' step in Theorem 1.1 is under-explained. The reader's worry about a O(1/T) phase gap is not right: the minimal nonzero gap is at least c/x, so the same Gaussian decay as in I1 applies. But the support property of r that makes it work should be stated. The placeholder '[?, Lemma 5]' is a real citation hole. And the F(c) vacuity is a load-bearing overstatement.

Net: the core f=1 result is probably correct and new; the paper as written overclaims badly. A serious referee could get it into shape after major revision, with Theorem 1.2 restated honestly. I'd engage with a revised version, not this one.

Referee Report

2 major / 3 minor

Summary. The paper studies large values of Dirichlet polynomials ∑_{n≤N} f(n)n^{it} for completely multiplicative coefficients with |f(n)|=1. Theorem 1.1 claims a lower bound in the transition range log N = (log T log_2 T)^{1/2} τ with τ=(log_2 T)^{O(1)} for all such f. Theorem 1.2 claims a stronger lower bound, with exponent √2√(log(T/N) log_3(T/N)/log_2(T/N)), in the range exp((log T)^{1/2+δ}) ≤ N ≤ √T for all f in a class F(c) defined by Re f(n) overline{f(m)} ≥ c for all m,n. The proofs use the resonance method, estimates from Hough's work, and GCD-sum bounds of de la Bretèche–Tenenbaum.

Significance. If Theorem 1.2 held for a genuinely large class of multiplicative functions, it would improve on the uniform result of Xu and Yang. Theorem 1.1, if fully proved, would also be a useful extension in the transition regime. The paper correctly identifies the relevant external tools (Hough's resonator estimates and GCD-sum bounds) and the overall strategy is coherent. However, the central generalization claim of Theorem 1.2 is empty, because the class F(c) contains only the constant function f≡1 (and is empty for c>1). This directly undermines the advertised improvement over Xu–Yang. The paper also leaves the key off-diagonal estimate in Theorem 1.1 at the level of an assertion.

major comments (2)

[Section 3, definition of F(c) and Eq. (3.8)] The class F(c) is essentially empty. For any prime p write f(p)=e^{iθ}. Since f is completely multiplicative, f(p^k)=f(p)^k. The condition Re f(n) overline{f(m)} ≥ c for all m,n, applied with m=1, gives Re f(p^k)=cos(kθ) ≥ c for every k≥1. If θ≠0 mod 2π, density (or rationality) gives some k with cos(kθ)<c. Hence θ=0 for every prime p, so f≡1. For c>1, F(c) is empty; for 0<c≤1, F(c)={1}. Thus Theorem 1.2 does not establish a result for a family of multiplicative functions; it reduces to f=1. The proof's step (3.8), which lower-bounds Re f(am) overline{f(bn)} by c, is vacuous beyond the constant function. This is a load-bearing error: the advertised improvement over Xu–Yang's uniform result is not realized.
[Section 2, after the expansion of I2] The sentence 'Similarly to our treatment for I1(R,T)' asserts the negligibility of the off-diagonal contribution to I2 without proof. In I1 the minimal logarithmic gap is c/x, giving a Gaussian argument cN/logT. In I2, when km≠n, the integers km and n differ by at least 1 and n≤x, so |log(km/n)| ≥ 1/(2x) and the Gaussian argument is at least N/(2 logT), which is large under the hypotheses. The step is therefore likely repairable. Nevertheless, the proof of Theorem 1.1 currently depends on an unverified claim at this point; a complete bound on the off-diagonal sum must be supplied.

minor comments (3)

[Section 3, proof of (3.3)] The citation 'similar to [?, Lemma 5]' is unresolved. Either replace it with a precise reference or remove it, since a proof is sketched.
[Throughout] There are numerous LaTeX corruption artifacts in the displayed equations (e.g., 'Nq 2 m/(m,n) n/(m,n)' and missing overlines in F(c)). These should be corrected in a revision.
[Theorem 1.1] The final step 'Theorem 1.1 follows from [11, p.105–107] immediately' is very terse. The exact estimate needed for ∑_{k≤N} r(k) should be stated explicitly so the reader can verify the constant A(τ+τ′).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central lower bounds are derived by the resonance method from external results (Hough; de la Bretèche–Tenenbaum); the only self-citation [7] is a baseline comparison and not load-bearing.

full rationale

The derivation chain is not circular. Theorem 1.1 constructs a resonator r from Hough's choice r(p)=... and reduces the second moment to sums Σ_{k≤N} r(k), citing Hough [11, pp. 105–107] for the final estimate; this is an external, independent source. Theorem 1.2 uses Lemma 3.1, quoted from de la Bretèche–Tenenbaum [6], and the rest is a standard resonance-method calculation. The self-citation [7] appears only in the sentence 'In particular, if we choose f(·)=1 always, then Theorem 1.1 overcomes Theorem 1.6 of [7]', i.e. as a comparative baseline, not as a proof ingredient. Two non-circular concerns should be noted. First, the class F(c) defined in Section 3 is almost certainly the singleton {f≡1}: Re f(p^k) ≥ c for all k forces each f(p)=1 by density/rational-order arguments, so the advertised uniformity over F(c) is vacuous and the claimed improvement over Xu–Yang [14] should be read as a result for f=1 only. This is a correctness/overclaim issue, not a circular reduction, because the f=1 bound is proven directly rather than imported from the definition. Second, the off-diagonal estimate in Theorem 1.1 is asserted with 'Similarly to our treatment for I1(R,T)' without the support argument needed to justify it, and Section 3 contains an incomplete citation '[?, Lemma 5]' for the M1 bound. These are gaps in exposition/proof, not instances of the paper's conclusions being equivalent to its inputs. Accordingly the circularity score is low; no step reduces by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters or new entities. It relies on established external theorems (Hough's sums, GCD sum lemma) and standard analytic number theory tools. The resonator construction is tailored to the problem but is a definition rather than an unproved assumption.

assumptions (5)

standard math Hough's estimate for the sum ∑_{k≤N} r(k) (Hough, Mathematika 59 (2013), pp. 105-107)
The final asymptotic in Theorem 1.1 is directly quoted from this external theorem; the proof does not re-derive it.
standard math GCD sum lemma (de la Bretèche and Tenenbaum, Proc. London Math. Soc. 119 (2019), Theorem 1.1)
Used in Lemma 3.1 to bound the maximal GCD sum over sets M with max M ≤ 2 min M.
standard math Standard resonance method and mean value estimates for Dirichlet polynomials (Montgomery–Vaughan)
Used implicitly in the moment computations and the lower bound for max |S_t| in terms of M2/M1.
domain assumption The set M in Theorem 1.2 satisfies max M ≤ 2 min M
This is part of the GCD sum lemma's condition; the proof relies on it for the inequality 1/max{u,v} ≥ 1/sqrt(2uv).
domain assumption The class F(c) condition: Re f(n) overline{f(m)} ≥ c for all m,n
The lower bound in the I2 expansion uses this to control the phase of off-diagonal terms in Theorem 1.2.

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Pith. "Pith review of Large values of Dirichlet polynomials with multiplicative coefficients." pith.science (2026). https://pith.science/paper/SURD76Q4

@misc{pith2026250909771,
  author       = {Pith},
  title        = {Pith review of: Large values of Dirichlet polynomials with multiplicative coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SURD76Q4}},
  note         = {Machine review of arXiv:2509.09771}
}

abstract

In this paper, we investigate large values of Dirichlet polynomials with multiplicative coefficients $\sum_{n\le N}f(n)n^{it}$, where $1\ll t\le T$ for large $T$. We prove an improved Omega result in the region $\exp((\log T)^{\frac12+\varepsilon})\le N\le\sqrt T$, where $T$ is large. We also show an Omega result when $\log N$ is around $\sqrt{\log T\log_2T}$.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

14 extracted references · 1 linked inside Pith

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [14]

W., Yang, D.Extreme values of Dirichlet polynomials with multiplicative coefficients,J

Xu, M. W., Yang, D.Extreme values of Dirichlet polynomials with multiplicative coefficients,J. Number Theory,258(2024), 173–180

2024

[2] [1]

Ann.,365(2016), 473–496

Aistleitner, C.Lower bounds for the maximum of the Riemann zeta function along vertical lines, Math. Ann.,365(2016), 473–496. LARGE V ALUES OF DIRICHLET POLYNOMIALS WITH MULTIPLICATIVE COEFFICIENTS 11

2016

[3] [2]

and Nishry, A.Extremal bounds for Dirichlet polynomials with random multiplicative coefficients,Stud

Benatar, J. and Nishry, A.Extremal bounds for Dirichlet polynomials with random multiplicative coefficients,Stud. Math.,2023, 59–80

2023

[4] [3]

Bondarenko, A., and Seip, K.Helson ’s problem for sums of a random multiplicative function, Mathematika62(2016), 101–110

2016

[5] [4]

J.,166(2017), 1685–1701

Bondarenko, A., Seip, K.Large greatest common divisor sums and extreme values of the Riemann zeta function,Duke Math. J.,166(2017), 1685–1701

2017

[6] [5]

de la Bret` eche, R., Munsch, M., and Tenenbaum, G.Small G´ al sums and applications,J. Lond. Math. Soc. (2),103(2021), no. 1, 336–352

2021

[7] [6]

de la Bret` eche, R., Tenenbaum, G.Sommes de G´ al et applications (in French) [G´ al-type sums and applications],Proc. Lond. Math. Soc.,119(2019), 104–134

2019

[8] [7]

Dong, Z., Wang, W., Zhang, H.Large zeta sums,Period. Math. Hung., 1–11