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Omega Results for The Divisor and Circle Problems Using The Resonance Method

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Pith's one-line read Sharper omega bounds for divisor and circle problems follow from a new resonance theorem.

desk verdict A plausible resonance-method framework whose intended Omega improvements are genuine, but the applications as written have a scaling mismatch that collapses the main term. read the letter →

arxiv 2504.17032 v2 pith:BLUBYO76 submitted 2025-04-23 math.NT

Kamalakshya Mahatab This is my paper · ORCID
classification math.NT MSC 11N3711M0611P21
keywords omegaresultsresonancemethodDirichletdivisorproblemGausscirclePiltzmeansquareofRiemannzetafunctionexponentialsumslowerbounds
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 aims to prove that the error terms in the Dirichlet divisor problem, the Gauss circle problem, the k-fold Piltz divisor problem, and the mean-square formulas for the Riemann zeta function and Dirichlet L-functions all oscillate more than earlier lower bounds showed. Its main theorem gives a resonance-method lower bound for general exponential sums with nonnegative coefficients, and the applications sharpen the triple-logarithm factor in Soundararajan's bounds from (log_3 x)^{-5/8} to (log_3 x)^{-3/8}, with the analogous improvement for the mean-square error terms over Lau and Tsang. A reader should care because these $\Omega$-results are the known lower limits on how far these error terms can deviate from their conjectured $x^{{1/4+epsilon}}$ size.

What carries the argument

The load-bearing object is the resonator $R(x)=\prod_{\lambda_n\in M}(1-e^{-\lambda_n/2\alpha}e^{i\lambda_n x})^{-1}$, built from a rationally independent set $M$ of frequencies in $[C_1\alpha,2\alpha]$. Its square integrates to about $\sqrt{2\pi}Y_2 e^{M/7}$ and is pointwise at most $e^{2M/C_1}$. The argument pairs $R$ with the convolution identity $\int F_\beta(x+u)(\sin\alpha u/u)^2 e^{-2i\alpha u}du = \frac12 e^{i\beta}\sum_{\lambda_n} a_n w_\alpha(\lambda_n)e^{i\lambda_n x}$, whose weight $w_\alpha(\lambda_n)=\frac{\pi}{2}\max\{0,2\alpha-|\lambda_n-2\alpha|\}$ suppresses all frequencies above $2\alpha$. Positivity of $a_n$ lets the proof discard off-diagonal terms in the double integral defining $I_1$, turning $|I_1|/|I_2|$ into $\frac12\sum_{\lambda_n\in M}a_n r(\lambda_n)w_\alpha(\lambda_n)$; since $r(\lambda_n)\ge e^{-1}$ and $w_\alpha\ge\pi\alpha$ on $M$, the main term is $\frac{\pi}{4e}\sum_{\lambda_n\in M}a_n$.

What would settle it

Take the divisor application's explicit value $\alpha$ = (1/C)(log X)(log_2 X)^{1 - $\lambda$ + $\lambda$ log $\lambda$}(log_3 X)^{1/2} for large X and compute the two sets {n : n in [C_1 $\alpha$, 2 $\alpha$]} and {n : 4 pi $\sqrt$(n) in [C_1 $\alpha$, 2 $\alpha$]}. The second condition forces n of size about $alpha^{2}$, while the first has n at most 2 $\alpha$, so for X large these intervals are disjoint; under the paper's literal notation the set M used in the divisor application is empty and the main term of Theorem 6 contributes nothing. Checking this one-line inequality settles whether the proof, as written, reaches the stated divisor bound.

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Extended reading notes

Core claim

At the center is Theorem 6, stated for $F_\beta(x)=\sum_{n\le X^{A_1}} a_n\cos(x\lambda_n+\beta)$ with $a_n\ge0$ and $\lambda_n>0$: if $M\subset\{\lambda_n:C_1\alpha\le\lambda_n\le 2\alpha\}$ is rationally independent and has size $M$, then $\max_{X^{A_3}/2<x\le 2^{A_2}X^{A_2}(\log X)^2}|F_\beta(x)|\ge \frac{\pi}{4e}\sum_{\lambda_n\in M}a_n$ up to controlled error terms. The applications take $M$ to be the frequencies attached to square-free integers in a short interval with about $\lambda\log_2\alpha$ prime factors, with a residue-class version for the circle problem. Feeding these choices into Voronoi-type series yields Theorem 1: $\max_{X/2<x\le 5X^{3/2}(\log X)^2}|\Delta(x^2)|/\sqrt{x}\gg(\log X)^{1/4}(\log_2 X)^{(3/4)(2^{4/3}-1)}(\log_3 X)^{-3/8}$, hence $\Delta(x)=\Omega((x\log x)^{1/4}(\log_2 x)^{(3/4)(2^{4/3}-1)}(\log_3 x)^{-3/8})$; the same pattern gives the corresponding theorems for the circle, Piltz, and mean-square error terms.

Load-bearing premise

The load-bearing premise is that one cut-off $\alpha$ controls both the frequency window lambda_n in [C_1 $\alpha$, 2 $\alpha$] and the integer window n in [C_1 $\alpha$, 2 $\alpha$], a double use that for the $\alpha$-values in the applications makes the two windows disjoint (the second forces n of size $alpha^{2}$, while the first allows only n up to 2 $\alpha$), and the main term of the lower bound depends on this identification.

Editorial extensions

If this is right

  • The Dirichlet divisor error is infinitely often at least $C\sqrt{x}(\log x)^{1/4}$ times the displayed iterated-logarithm factors, so the true size of $\Delta(x)$ is strictly above the $x^{1/4}$ scale by a growing factor.
  • The same growth rate holds for the Gauss circle error $P(x)$ and for the normalized mean-square errors $E(2\pi x^2)/\sqrt{x}$ and $E(q,2\pi x^2)/\sqrt{x}$.
  • For each $k\ge2$, the Piltz divisor error has the analogous lower bound with the iterated-log exponents stated in Theorem 3, including the sign-only versions when $k\equiv3$ or $7\pmod8$.
  • The error terms in the resonance lower bound are $O(X^{-1/8})$ in the divisor and circle applications, so the lower bounds survive the truncation and smoothing steps used to pass from the Voronoi series to the exponential-sum model.

Reading between the lines

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

  • Because the literal indexing in the applications makes the upheld frequency set empty, a sympathetic repair would define the resonator over integers $n$ with $4\pi\sqrt{n}\in[C_1\alpha,2\alpha]$; the Sathe-type count would then run over integers of size roughly $\alpha^2$, and the optimization of $\alpha$ would need to be redone.
  • The positivity of the coefficients is doing essential work: it lets the proof discard all off-diagonal pairs and keep a nonnegative main term. The theorem, as stated, does not directly handle signed sums without folding a sign pattern into the resonator coefficients.
  • If the stronger Sathe-type lower bound sketched in the introduction, for integers in $[\alpha,C\alpha]$ with more than $\lambda\log_2\alpha$ prime factors, can be proved, the factor $(\log_3 x)^{-3/8}$ would be removable and the lower bounds would reach the conjecturally optimal logarithmic shape.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper proposes to use the resonance method to prove Omega results for the Dirichlet divisor problem, the circle problem, the Piltz divisor problem, and the mean-square error terms for the Riemann zeta function and for Dirichlet L-functions. The central tool is Theorem 6, which gives a lower bound for a general exponential sum with positive coefficients, and the applications in Section 4 aim to improve Soundararajan's known exponents by replacing (log_3 x)^{-5/8} with (log_3 x)^{-3/8} in several settings. The paper is clearly structured, with a general convolution lemma, a resonator construction, and explicit error estimates.

Significance. If the claimed results were correct, they would constitute a genuine improvement over Soundararajan's 2003 Omega bounds and over the Lau-Tsang mean-square results, representing a meaningful step in the study of large values of divisor and lattice-point error terms. The paper also articulates a useful general framework for applying the resonance method to exponential sums with positive coefficients. However, the central derivation contains a scaling inconsistency between the resonator set defined in the applications and the hypothesis of Theorem 6, and the main-term lower bound in Theorem 6 relies on an incorrect inequality for the convolution weight. These issues are load-bearing and currently leave the main theorems unsupported.

major comments (2)
  1. [Section 4.1, applied uniformly in Sections 4.2-4.5] The resonator set used in the applications is incompatible with the hypothesis of Theorem 6. In Section 4.1, the set is defined as M = {lambda_n : n in [C1 alpha, 2 alpha], omega(n) = [lambda log_2 alpha], n squarefree}, with lambda_n = 4 pi sqrt(n). Theorem 6, however, requires M to be contained in {lambda_n : C1 alpha <= lambda_n <= 2 alpha}. Since lambda_n = 4 pi sqrt(n), the condition n in [C1 alpha, 2 alpha] gives lambda_n of size sqrt(alpha), not alpha, for large alpha; conversely, the condition lambda_n in [C1 alpha, 2 alpha] would force n to be of size alpha^2. Thus the set M as written does not satisfy the hypothesis of Theorem 6, so the main-term lower bound of Theorem 6 cannot be applied. If n is interpreted literally as being of size alpha, then w_alpha(lambda_n) = (pi/2) lambda_n is of size sqrt(alpha), and the main term in Theorem 6 would be smaller than claimed by a factor of alpha^{-1/2}. If the construction is instead corrected to make lambda_n of size alpha, then n is of size alpha^2 and the count and sums in Section 4.1 change, and the stated lower bound is not recovered. The same mismatch appears in the circle problem, the k-divisor problem, and the mean-square applications, all of which use the same M-construction.
  2. [Section 3, proof of Theorem 6] The proof of Theorem 6 uses the inequality w_alpha(lambda_n) >= alpha pi for all lambda_n in M. According to Lemma 1, w_alpha(lambda) = (pi/2) max{0, 2 alpha - |lambda - 2 alpha|}. For 0 < lambda <= 2 alpha this simplifies to w_alpha(lambda) = (pi/2) lambda. Therefore w_alpha(lambda_n) >= alpha pi would require lambda_n >= 2 alpha, which is only possible at the endpoint of the interval [C1 alpha, 2 alpha] and is false throughout the interior for any C1 < 2. Since this inequality is used to pass from (1/(4 alpha)) sum_{lambda_n in M} a_n r(lambda_n) w_alpha(lambda_n) to the claimed (pi/(4e)) sum_{lambda_n in M} a_n, the main-term constant in Theorem 6 is not justified. A corrected bound of the form w_alpha(lambda_n) >= (pi/2) C1 alpha would change the constant but would not repair the scaling mismatch described in the previous comment.
minor comments (3)
  1. [Theorem 6 statement] The displayed range for the maximum is garbled: "max_{X^{A3}/2<x<=2A2 2X^{A2}(logX)^2}" should read "max_{X^{A3}/2 < x <= 2X^{A2}(log X)^2}".
  2. [Section 4.4, equation (6)] The notation "a_j b_-i" appears to contain a typo; the product should presumably be "a_j b_{-i}" or similar, matching the definitions of the constants a and b just above the equation.
  3. [Section 2.2, footnote 2] The remark that each lambda_n has r(lambda_n) < 1 and the resonator is extended multiplicatively is helpful, but it would be clearer to state explicitly that this multiplicative extension is what makes the Euler product in equation (1) valid.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity found: the claimed Omega-bounds are derived from a general resonance lower bound with explicitly constructed resonator sets; the flagged frequency-window mismatch is a correctness defect, not a circular reduction.

full rationale

The paper's derivation chain is self-contained in the sense required by the circularity test. Theorem 6 gives a lower bound for a general exponential sum in terms of an explicitly chosen resonator set M, and the applications compute the resulting main term using independent external ingredients: Voronoi summation, Sathe's theorem, Tenenbaum's estimates, and the Lau–Tsang identities. The parameters alpha and lambda are optimized inside the derivation, and the final log_3 x exponent emerges from that optimization; no parameter is fitted to the target Omega-bound and then renamed as a prediction. The only self-citation, [2] by Aistleitner–Mahatab–Munsch, is used as inspiration for the resonator construction and for the positivity technique in Lemma 3, but the proof of Lemma 3 is written out and does not import the conclusion from [2]. No uniqueness theorem from the authors' prior work is invoked, and no known result is merely renamed. The serious issue raised in the reading notes—that the resonator set in Section 4.1 is defined by n in [C_1 alpha, 2 alpha] while Theorem 6 requires lambda_n in [C_1 alpha, 2 alpha] with lambda_n = 4 pi sqrt(n)—is an internal scaling inconsistency that would undermine the main-term lower bound as written. That is a mathematical correctness defect, not a circularity, because the lower bound is not assumed from the target result and the proof does not reduce to an identity between input and output. Accordingly, the circularity score is low.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard analytic number theory inputs (Voronoi, Sathe, Stirling, linear independence of square roots) plus the paper's resonator construction. The main unresolved burden is not in background facts but in the internal consistency of the resonator set: the applications define M by a condition on n while Theorem 6 requires a condition on lambda_n. Several implied quantities are not specified.

free parameters (4)
  • alpha (spectral scale) = (1/C)(log X)(log_2 X)^{1-lambda+lambda log lambda}(log_3 X)^{1/2}
    Chosen to balance the main term against error terms in the resonance estimate; the final exponent depends on this choice.
  • lambda (prime-factor parameter) = 2^{4/3} for divisor, circle, and zeta applications; k^{2k/(k+1)} for Piltz
    Variational parameter controlling the number of prime factors in the resonator set; optimized to maximize the lower bound, not fitted to external data.
  • C (large constant in alpha) = unspecified, chosen large
    Appears in alpha and is used to force e^{2M/C1} to be small enough; no explicit value is given.
  • C1 (frequency window endpoint) = 0 < C1 < 2, unspecified
    Defines the lower endpoint of the spectral window for M; affects constants in the main-term lower bound.
assumptions (5)
  • domain assumption The truncated Voronoi identity quoted in Section 1 extends uniformly to the wider interval used in the theorems, up to 5 X^{3/2}(log X)^2.
    The formula as quoted from [16] is stated for sqrt(X) <= x <= X^{3/2}, while Theorems 1-3 maximize over a wider interval.
  • standard math Sathe's theorem gives the stated asymptotic for the number of squarefree integers with a prescribed number of prime factors.
    Used in Section 4.1 to estimate the size of the resonator set M.
  • standard math Voronoi summation gives the truncated expansion of the divisor and circle error terms with the stated error term.
    Invoked in Sections 1 and 4.1 to convert the resonance lower bound into Omega results.
  • standard math Square roots of distinct squarefree integers are linearly independent over Q.
    Needed so that the resonator R(x) has the Euler product form and the diagonal terms in I2 dominate.
  • domain assumption The Lau-Tsang identities in Section 4.4 allow the mean-square zeta error to be reduced to the exponential sum Q(x,tau) with the claimed linear combination of P terms.
    The paper imports these identities and then asserts, without proof, that the resonance calculation can be carried out for Q(x,tau).

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Cite this review

Pith. "Pith review of Omega Results for The Divisor and Circle Problems Using The Resonance Method." pith.science (2026). https://pith.science/paper/BLUBYO76

@misc{pith2026250417032,
  author       = {Pith},
  title        = {Pith review of: Omega Results for The Divisor and Circle Problems Using The Resonance Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLUBYO76}},
  note         = {Machine review of arXiv:2504.17032}
}
abstract

We apply the resonance method to obtain large values of general exponential sums with positive coefficients. As applications, we show improved $\Omega$-bounds for Dirichlet and Piltz divisor problems, Gauss circle Problem, and error term for the mean square of the Riemann zeta function and the Dirichlet $L$-function.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Piltz divisor Problem in Number Fields Using The Resonance Method

    math.NT 2025-06 conditional novelty 6.0 of 10

    For the Piltz divisor problem over number fields, the paper proves an improved Omega lower bound with a larger third-log exponent than previous results.

  2. Omega Estimate for the Lattice Point Discrepancy of a Body of Revolution Using The Resonance Method

    math.NT 2025-06 conditional novelty 5.0 of 10

    For smooth convex bodies of revolution in R^3, the lattice point discrepancy satisfies P_B(t) = Omega_-(t^(1/2)(log t)^(1/3)(log_2 t)^((2/3)(sqrt(2)-1))(log_3 t)^(-1/3)), improving the prior exponent of log_3 t from -...

Reference graph

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