The Piltz divisor Problem in Number Fields Using The Resonance Method

Kamalakshya Mahatab; Nilmoni Karak

REVIEW 3 major objections 5 minor 15 references

The Piltz divisor Problem in Number Fields Using The Resonance Method

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For every number field, the Piltz divisor error term obeys a sharper $\Omega$ lower bound with the third iterated-logarithm exponent raised by $(mk-1)/(2mk)$.

desk verdict A genuinely new Omega bound for the Piltz divisor problem over number fields, but the key Voronoi-type formula needs cleanup before the proof is checkable. read the letter →

arxiv 2506.22587 v1 pith:USDPZDMF submitted 2025-06-27 math.NT

Nilmoni Karak , Kamalakshya Mahatab This is my paper

classification math.NT MSC 11R4211P2111N37

keywords PiltzdivisorproblemnumberfieldsOmegaresultsresonancemethodVoronoisummationDedekindzetafunctionDirichletGausscircle

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 proves an improved lower-bound statement for the error in counting $k$-tuples of integral ideals of bounded norm in any algebraic number field. Concretely, for a field of degree $m$ and any positive integer $k$ with $mk\ge 2$, the error $\Delta_K^{(k)}(x)$ exceeds $(x\log x)^{(mk-1)/(2mk)}(\log_2 x)^\beta(\log_3 x)^\gamma$ infinitely often, where $\beta$ is fixed by the prime-splitting densities of $K$ and $\gamma$ is determined by the number $R$ of splitting types with positive density. This improves the best prior lower bound by raising the exponent of the third logarithm by $(mk-1)/(2mk)$. The proof combines a Voronoi-type summation formula, which turns the smoothed error into a series of cosine waves, with the resonance method, which engineers a set of integers whose large divisor counts force one of those waves to be large. The theorem includes the classical Dirichlet divisor problem and the Gauss circle problem as special cases, and it yields sign-specific $\Omega_+$ or $\Omega_-$ results when $kr_1$ is $3$ or $7$ modulo $8$.

What carries the argument

The core ingredient is a Voronoi-type summation formula for number fields (Proposition 2.2): after smoothing with a Gaussian kernel, $\Delta_K^{(k)}(x)$ becomes a series over $n$ of cosine terms with amplitude $D^{1/(2m)}/(\pi\sqrt{mk})\,x^{(mk-1)/2}\,d_K^{(k)}(n)n^{-(mk+1)/(2mk)}$, exponential factor $e^{-\pi^2(n/(D^k\alpha))^{2/(mk)}}$, and phase $2\pi mk x(n/D^k)^{1/(mk)}+\pi(kr_1-3)/4$, plus a power-saving error. The resonance method enters through Theorem 2.1, which guarantees that the maximum over a short interval of $x$ of such a cosine sum is at least a constant multiple of the total weight of a carefully chosen finite set $M$ of squarefree integers, provided their frequencies are linearly independent over $\mathbb{Q}$. The set $M$ is built so that each of its elements has many prime factors from the splitting classes $P_\nu$, which makes $d_K^{(k)}(n)$ large and makes the counting of $M$ depend on the Dirichlet densities $\delta_\nu$. The stationary-phase evaluation of the Voronoi integral is what produces the phase $\pi(kr_1-3)/4$, and that phase is exactly what later determines the $\Omega_+$/$\Omega_-$ refinement.

What would settle it

Compute the left side of Proposition 2.2 numerically for a concrete case such as $K=\mathbb{Q}$, $k=2$, using the exact Perron-integral form in (2.3)--(2.4), and compare it with the right-hand cosine series; a mismatch larger than the claimed $O(x^{mk/2-3/5}\alpha^{1/2+\epsilon})$ error would disprove the key step.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for every algebraic number field $K$ of degree $m=r_1+2r_2$ and every positive integer $k$ with $mk\ge 2$, the Piltz divisor error term $$\$Delta_K^{{(k)}}$(x)=\$\Omega$\!\left((x\log x)^{\frac{mk-1}{2mk}}(\log_2 x)^\$\beta$(\log_3 x)^\gamma\right)$$ holds as $x\to\infty$, with $\beta$ the density-weighted constant in (1.7) and $\gamma=-(mk+1)R/(4mk)$, where $R$ is the number of splitting types $\nu$ with $\delta_\nu>0$. Here $\log_2 x$ and $\log_3 x$ are the second and third iterated logarithms. This improves the previous bound of the same shape by raising the exponent of $\log_3 x$ by $(mk-1)/(2mk)$. Moreover, when $kr_1\equiv 3\pmod 8$ the lower bound can be taken as $\Omega_+$ (positive infinitely often), and when $kr_1\equiv 7\pmod 8$ as $\Omega_-$ (negative infinitely often).

Load-bearing premise

The proof stands or falls on Proposition 2.2's identity expressing the smoothed error as a cosine series (reading the evident typo in (2.2), $e^{u/s}$, as $e^{u/x}$): if its phase, amplitude, or error estimate is off, the final exponents change.

Editorial extensions

If this is right

For the classical Dirichlet divisor problem ($K=\mathbb{Q}$, $k=2$) and the Gauss circle problem ($K=\mathbb{Q}(i)$, $k=1$), the theorem recovers the sharpest current lower bounds, with third-log exponent $-3/8$.
For a normal extension of degree $m\ge 2$, the result simplifies to an explicit formula with third-log exponent $-(mk+1)/(4mk)$, improving the previous bound by $(mk-1)/(2mk)$.
When $kr_1\equiv 3\pmod 8$, the error term is positive infinitely often; when $kr_1\equiv 7\pmod 8$, it is negative infinitely often.
For the non-normal cubic field of Example 4.3 and the quintic field of Example 4.4, the theorem supplies fully explicit constants $\beta$ and $\gamma$, giving the first worked bounds for those fields with the improved third-log exponent.

Reading between the lines

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

Editorial inference: the same resonance-plus-Voronoi mechanism should extend to other zeta or $L$-functions whose functional equations have Gamma factors; the phase $\pi(kr_1-3)/4$ would become an archimedean parameter, so the sign-specific congruence would be a condition on the Hodge-type invariants rather than on $r_1$.
Editorial inference: the proof suggests that the size of the Piltz error is governed by the whole splitting profile of primes in $K$, not just by the degree $m$; fields with more splitting classes have a larger $R$ and therefore a steeper negative power of $\log_3 x$ in the lower bound.
Editorial inference: a numerical check of Proposition 2.2 for $K=\mathbb{Q}$, $k=2$ would isolate the single compressed step in the proof, since the rest of the argument is a counting estimate for the resonator set $M$.

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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

The paper does something real: it removes the (mk-1)/(2mk) term from the third-log exponent in the number-field Piltz Omega bound, extending the second author's rational-case result to arbitrary number fields. The main theorem is new, and the examples in Section 4 are useful. The overall strategy is sound: apply a recent resonance-method theorem from a companion paper, prove a number-field Voronoi-type formula, and set up the resonator using prime-splitting densities. The final exponents in Theorem 1.1 are explicit and match the expected generalization.

The soft spot is Proposition 2.2. The stationary-phase evaluation is the load-bearing step and it is compressed. The stress-test note correctly identifies a real discrepancy: the displayed (2.6) gives an amplitude (x n^{1/mk})^{mk/2-1}, which after summing against d(n)/n would give a different power of x than the claimed (2.1). A direct saddle-point computation from (2.4) yields the (mk/2 - 1/2) exponent, so (2.1) is likely right and (2.6) has a typo. That is repairable, but the text as written forces the reader to redo the calculation to verify. There are also smaller typos: e^{u/s} in (2.2) and (2.3) should be e^{u/x}, and (mν) in (3.4) should be (kν). More annoying is the overloaded use of α: in Theorem 2.1 it is the scale of λ_n, which corresponds to n ~ α^{mk}, while in Proposition 2.2 it is the smoothing parameter. The paper switches between these without comment, making (3.5) look wrong until you disentangle them. These are presentation issues, not fatal flaws.

The paper also quotes Theorem 2.1 from the second author's companion paper rather than proving it. That is acceptable if the companion paper is solid, but it limits self-containedness and should be flagged by the referee.

Who should read this? Anyone working on Omega estimates for divisor problems or applying the resonance method in number fields. The result is worth knowing. I would send it to a serious referee. The referee should focus on Proposition 2.2 and ask for a corrected, expanded stationary-phase argument, a clean separation of the two α scales, and a precise reference for Theorem 2.1. The central claim is very likely correct; it just needs a proper write-up.

Referee Report

3 major / 5 minor

Summary. The paper claims an improved Omega-bound for the Piltz divisor problem over number fields. For any number field K of degree m and any positive integer k with mk >= 2, Theorem 1.1 asserts that the error term Delta_K^{(k)}(x) is Omega((x log x)^{(mk-1)/(2mk)} (log_2 x)^beta (log_3 x)^gamma), with beta as in (1.7) and gamma = -(mk+1)/(4mk) R. This raises the exponent of log_3 x compared with the earlier bound of Girstmair et al. by removing the term -(mk-1)/(2mk). The proof combines a resonance theorem (Theorem 2.1) quoted from a companion paper by the second author with a new Voronoi-type formula for number fields (Proposition 2.2), and it also yields Omega_plus/Omega_minus refinements under congruences on kr1.

Significance. If the main theorem is correct, it is a genuine and meaningful improvement over the existing lower bounds for the Piltz divisor problem in number fields, generalizing the resonance method of the second author from the classical divisor and circle problems to arbitrary number fields. The Voronoi-type formula in Proposition 2.2 is potentially useful beyond this paper, and the manuscript is organized in a way that makes the extension to number fields transparent. The result is falsifiable and specific, and the claimed improvement is quantitative. However, the proof as written contains several gaps and notational inconsistencies in load-bearing steps, so the contribution is conditional on repair.

major comments (3)

[Section 2, equations (2.5)-(2.6)] The stationary-phase evaluation in the proof of Proposition 2.2 is not correct as displayed. In (2.5), the quadratic phase has coefficient of magnitude mk/(2t0), but the displayed coefficient is mk/(4πk)(D^k/n)^{1/mk}, missing the factor 1/x. More importantly, the Gaussian integral in (2.5) contributes a factor sqrt(π/a) of size x^{1/2} n^{1/(2mk)} D^{-1/(2m)}. Without this factor, the amplitude in (2.6) is (x n^{1/(mk)})^{mk/2-1}, whereas the correct stationary-phase amplitude is (x n^{1/(mk)})^{mk/2-1/2}. As written, the passage from (2.6) to the final formula (2.1) is inconsistent: multiplying (2.6) by d(n)/n gives x^{mk/2-1} n^{-1/2-1/mk}, whereas (2.1) has x^{(mk-1)/2} n^{-(mk+1)/(2mk)}. Since all subsequent Omega exponents are read off from (2.1), this step must be corrected and the missing Gaussian-integral evaluation supplied.
[Section 3, application of Theorem 2.1] The symbol alpha is used both for the Gaussian smoothing parameter in Proposition 2.2 and for the frequency scale in Theorem 2.1. In Section 3, the set M is constructed so that n is of size alpha, which is necessary for the exponential factor exp(-π^2(n/(D^k alpha))^{2/(mk)}) in (3.1) to be O(1). But with lambda_n = 2πmk(n/D^k)^{1/(mk)}, such n give lambda_n of size alpha^{1/(mk)}, not alpha. Theorem 2.1, however, is stated with lambda_n in [C1 alpha, 2 alpha]. Thus, as written, the resonance theorem is being applied with the wrong parameter. The proof should introduce a separate frequency-scale parameter, say beta = alpha^{1/(mk)}, restate the range of M in terms of beta, and check the error-term estimates (3.5)-(3.6) under that scaling. The final lower bound is likely repairable, but the current text is internally inconsistent on this point.
[Section 3, passage from (3.3) to (3.1)] The proof uses the lower bound from Theorem 2.1 on max_x |Σ f(n) cos(lambda_n x + theta)| and then, in (3.1), needs a lower bound on the signed sum Σ f(n) cos(lambda_n x + theta) itself. Since Theorem 2.1 is valid for any theta, applying it to both theta and theta + π gives an x for which the signed sum is positive and of the same order as the absolute maximum. This one-line argument should be stated explicitly; without it, the inference from the absolute-value bound to the lower bound for Delta_K^{(k)} is not justified.

minor comments (5)

[Section 2, equations (2.2)-(2.3)] In equations (2.2) and (2.3), the argument x^{mk} e^{u/s} should read x^{mk} e^{u/x}; the variable s is reused for the complex integration variable and creates confusion.
[Proposition 2.2, equation (2.1)] The error term in (2.1) is written as O(t^{mk/2-3/5} alpha^{2/(mk)}) with an undefined variable t; it should be x, and the exponent of alpha should be made consistent with the proof, where it appears as 1/2 + epsilon.
[Section 3, construction of M] The displayed range for the resonator set M, typeset as n in [C_1^{mk} alpha, 2mk alpha], is typographically ambiguous; it should be written unambiguously (for example as [C_1^{mk} alpha, 2^{mk} alpha]) and the exact relation to the condition on lambda_n in Theorem 2.1 should be spelled out.
[Lemma 2.3] The lemma uses the bound Delta_K^{(k)}(y) = O(y), which is not valid for k >= 2 since the trivial bound is O(y log^{k-1} y). The tail estimate still works after absorbing the logarithmic factor into the Gaussian decay, but the stated bound should be corrected.
[Throughout] There are several typographical errors, including 'PIL TZ' in the title, 'Vornoi' in the text before Proposition 2.2, and inconsistent use of 2km versus 2mk in the exponent n^{-(mk+1)/(2km)} near the end of the proof of Proposition 2.2.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the proof is derived from a general resonance theorem and an in-paper Voronoi-type formula; the load-bearing self-citation is not definitionally circular.

full rationale

The derivation chain is not circular. The final Omega-exponent is obtained by combining Proposition 2.2, a Voronoi-type formula proved in the paper via the functional equation and stationary phase, with Theorem 2.1, quoted from the second author's companion paper [8], which supplies a lower bound for a weighted cosine sum. The target statement about Delta_K^{(k)}(x) does not appear in the assumptions or conclusion of Theorem 2.1; that theorem is a general maximal inequality with free parameters f(n), lambda_n, alpha, A_i, and its conclusion concerns a trigonometric sum, not the divisor error term. Proposition 2.2 is proved in situ, and the final resonator-sum estimates use Dirichlet densities only to choose the set M and to optimize the exponent mu_nu. The optimization of mu_nu is a variational calculation, not data fitting. Thus no 'prediction' reduces by construction to its input. The only self-citation is load-bearing, but it is an independent general lemma whose stated assumptions do not include the target result, so under the stated rules it is not circular. The in-line inconsistencies in the stationary-phase proof, such as e^{u/s} for e^{u/x} in (2.2) and the displayed power of x n^{1/mk} in (2.6), are correctness/typo risks rather than circularity, since they do not make the conclusion definitionally equivalent to an input. Score 2 reflects the self-citation dependency without treating it as circular.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new particles, forces, or algebraic objects beyond standard ideals, prime-splitting types, and the resonator set. The free parameters are variational and are optimized away; the axioms are the resonance theorem from a companion paper, the stationary-phase asymptotic, and standard analytic number theory inputs.

free parameters (2)

mu_nu (resonator weights per prime-splitting type) = delta_nu (k nu)^(2mk/(mk+1))
Introduced in the construction of the resonator set M and optimized to maximize the power of log_2 X. They are variational parameters, not empirical fits, and they are eliminated in the final bound.
C(k) (small constant in the choice of alpha) = small positive constant
Chosen small enough so that e^(2M/C1) is negligible compared to X^(1/4-epsilon). It does not enter the final exponents.

assumptions (4)

domain assumption Theorem 2.1 (Mahatab [8])
The resonance lower bound for absolute cosine sums is imported from a companion paper by the second author and not proved in this paper.
ad hoc to paper The Voronoi-type asymptotic (2.6) from stationary phase
The claimed leading term and error in Proposition 2.2 rest on a compressed stationary-phase evaluation, which is asserted without a full derivation.
domain assumption Q-linear independence of {2 pi mk (n/D^k)^(1/(mk)) : n in M}
This hypothesis is required to apply Theorem 2.1. The paper asserts M is a set of squarefree integers but does not explicitly prove the linear independence of the corresponding lambda_n.
domain assumption The count M asyomped alpha (log alpha)^kappa (log_2 alpha)^(-mu) from [2]
The size of the resonator set is taken from the proof of Theorem 1 of Girstmair et al. and is not re-derived here.

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Cite this review

Pith. "Pith review of The Piltz divisor Problem in Number Fields Using The Resonance Method." pith.science (2026). https://pith.science/paper/USDPZDMF

@misc{pith2026250622587,
  author       = {Pith},
  title        = {Pith review of: The Piltz divisor Problem in Number Fields Using The Resonance Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/USDPZDMF}},
  note         = {Machine review of arXiv:2506.22587}
}

abstract

The Piltz divisor problem is a natural generalization of the classical Dirichlet divisor problem. In this paper, we study this problem over number fields and obtain improved $\Omega-$bounds for its error terms. Our approach involves generalizing a Voronoi-type formula due to Soundararajan in the number field setting, and applying a recent result due to the second author.

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Reference graph

Works this paper leans on

15 extracted references · 12 canonical work pages

[1]

Aistleitner, K

C. Aistleitner, K. Mahatab and M. Munsch, Extreme values of the Riemann zeta function on the 1-line, Int. Math. Res. Not. 2019, no. 22, 6924–6932

work page 2019
[2]

Girstmair, and M

K. Girstmair, and M. K¨ uhleitner, and W. M¨ uller, and W. G. Nowak, The Piltz divisor problem in number fields: an improved lower bound by Soundararajan’s method, Acta Arith. 117 (2005), no. 2, 187–206

work page 2005
[3]

J. L. Hafner, On the average order of a class of arithmetical functions, J. Number Theory 15 (1982), 36–76

work page 1982
[4]

J. L. Hafner, The distribution and average order of the coefficients of Dedekind ζ functions, J. Number Theory 17 (1983), no. 2, 183–190

work page 1983
[5]

M. N. Huxley, Exponential sums and lattice points. II, Proc. London Math. Soc. (3) 66 (1993), no. 2, 279–301

work page 1993
[6]

M. N. Huxley, Exponential sums and lattice points. III, Proc. London Math. Soc. (3) 87 (2003), no. 3, 591–609

work page 2003
[7]

Li and X

X. Li and X. Yang, An improvement on Gauss’s Circle Problem and Dirichlet’s Divisor Problem, arXiv, 2025, arXiv: 2308.14859

arXiv 2025
[8]

Mahatab, Omega Results for The Divisor and Circle Problems Using The Resonance Method, arXiv, 2025, arXiv: 2504.17032

K. Mahatab, Omega Results for The Divisor and Circle Problems Using The Resonance Method, arXiv, 2025, arXiv: 2504.17032

arXiv 2025

Show all 15 references

[9]

W. G. Nowak, On the distribution of integer ideals in algebraic number fields, Math. Nachr. 161 (1993), 59–74

1993
[10]

W. G. Nowak, On the general asymmetric divisor problem, Abh. Math. Sem. Univ. Hamburg 65 (1995), 259–276

1995
[11]

Soundararajan, Omega results for the divisor and circle problems, Int

K. Soundararajan, Omega results for the divisor and circle problems, Int. Math. Res. Not. 2003, no. 36, 1987–1998

2003
[12]

Szeg¨ o and A

G. Szeg¨ o and A. Walfisz,¨Uber das Piltzsche Teilerproblem in algebraischen Zahlk¨ orpern, Math. Z.26 (1927), no. 1, 138–156

1927
[13]

Szeg¨ o and A

G. Szeg¨ o and A. Walfisz,¨Uber das Piltzsche Teilerproblem in algebraischen Zahlk¨ orpern, Math. Z.26 (1927), no. 1, 467–486

1927
[14]

Takeda, Uniform bounds of the Piltz divisor problem over number fields, Pacific J

W. Takeda, Uniform bounds of the Piltz divisor problem over number fields, Pacific J. Math. 301 (2019), no. 2, 601–616

2019
[15]

E. C. Titchmarsh, The theory of the Riemann zeta-function , second edition, Oxford Univ. Press, New York, 1986. Nilmoni Karak, Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur- 721302, India. Email address : nilmonikarak@gmail.com, nilmonimath@kgp...

1986

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Aistleitner, K

C. Aistleitner, K. Mahatab and M. Munsch, Extreme values of the Riemann zeta function on the 1-line, Int. Math. Res. Not. 2019, no. 22, 6924–6932

work page 2019

[2] [2]

Girstmair, and M

K. Girstmair, and M. K¨ uhleitner, and W. M¨ uller, and W. G. Nowak, The Piltz divisor problem in number fields: an improved lower bound by Soundararajan’s method, Acta Arith. 117 (2005), no. 2, 187–206

work page 2005

[3] [3]

J. L. Hafner, On the average order of a class of arithmetical functions, J. Number Theory 15 (1982), 36–76

work page 1982

[4] [4]

J. L. Hafner, The distribution and average order of the coefficients of Dedekind ζ functions, J. Number Theory 17 (1983), no. 2, 183–190

work page 1983

[5] [5]

M. N. Huxley, Exponential sums and lattice points. II, Proc. London Math. Soc. (3) 66 (1993), no. 2, 279–301

work page 1993

[6] [6]

M. N. Huxley, Exponential sums and lattice points. III, Proc. London Math. Soc. (3) 87 (2003), no. 3, 591–609

work page 2003

[7] [7]

Li and X

X. Li and X. Yang, An improvement on Gauss’s Circle Problem and Dirichlet’s Divisor Problem, arXiv, 2025, arXiv: 2308.14859

arXiv 2025

[8] [8]

Mahatab, Omega Results for The Divisor and Circle Problems Using The Resonance Method, arXiv, 2025, arXiv: 2504.17032

K. Mahatab, Omega Results for The Divisor and Circle Problems Using The Resonance Method, arXiv, 2025, arXiv: 2504.17032

arXiv 2025