{"id":"ff17812d-a126-48c7-b7ff-b5280f04f91b","arxiv_id":"2504.17032","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"Improved logarithmic factors in Omega bounds for divisor, circle, Piltz, and zeta mean-square error terms are claimed via a resonance argument, but key proof steps are inconsistent or omitted.","lead":"This paper claims improved Omega lower bounds for classical error terms in the divisor, circle, Piltz divisor, and zeta mean-square problems, using a resonance construction for exponential sums. The main theorem is plausible, but the written proof contains scaling inconsistencies and several unproved steps, so the improvements are not established as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Scaling mismatch between the resonator set and the spectral window in Theorem 6 invalidates the main-term lower bound in the applications.","rationale":"After reading the full text, the decisive obstruction is the scaling of the resonator set in the applications. The general theorem is internally set up with λ_n as the spectral variable, and its proof requires λ_n≈α for the triangular weight w_α to be ≈α. But every application defines M through an interval for n, not for λ_n. For the divisor problem λ_n=4π√n and n∈[C1α,2α], so λ_n≈√α; the convolution window in Lemma 1 is centered at 2α, so the selected frequencies contribute only w_α≈√α. This is not a technical slip in a constant: replacing w_α by its actual size changes the main-term exponent. The same problem recurs in the circle and Piltz cases, where λ_n=2π√n and λ_n=2πk n^{1/k} respectively. I agree with the reader that this is the weakest assumption. I would not change the verdict: REJECT for the preprint as written. A repair would require either redefining M as {λ_n: λ_n∈[C1α,2α]} (so n≈α²) and recomputing all the sums, or reworking the main theorem with a spectral window matched to λ_n≈√α, and it is not clear that the claimed exponents would survive.","tokens_in":12559,"tokens_out":7864,"duration_ms":71030,"concrete_test":"A direct calculation settles the issue. Fix C1=1 and take α=10^m. Define M_literal={n∈[α,2α]: n squarefree, ω(n)=λ log_2 α} and M_spectral={n∈[α²,2α²]: n squarefree, ω(n)=λ log_2(α²)}. For each, compute S(M)=(1/(4α)) Σ_{n∈M} d(n)n^{-3/4} w_α(4π√n) r(4π√n) with w_α and r as in Lemmas 1 and §2.2, using the α chosen in §4.1. If S(M_literal) is O(α^{-1/4} (log α)^{O(1)}) while the claimed lower bound is ≫α^{1/4}(log α)^{...}, the proof's main-term step fails. Equivalently, re-derive the final displayed lower bound of §4.1 with w_α(λ_n)=πλ_n instead of απ; if the resulting Ω bound loses the (log X)^{1/4} factor, Theorem 1 does not follow from the written proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central lower bound in Theorem 6 is π/(4e) Σ_{λ_n∈M} a_n. Its proof uses r(λ_n)≥e^{-1} and w_α(λ_n)≥απ. But w_α(λ)=π max{0,2α−|λ−2α|}; for 0<λ≤2α this equals πλ, so w_α(λ_n)≥απ only when λ_n≥α. The hypothesis of Theorem 6 is λ_n∈[C1α,2α], which can satisfy this if C1≥1. However, in §4.1 the resonator set is defined as M={λ_n: n∈[C1α,2α], ω(n)=[λ log2α], n squarefree}, with λ_n=4π√n. Thus λ_n≈4π√α, not α. Consequently w_α(λ_n)=πλ_n≈√α, and the main term in Theorem 6 is smaller than claimed by a factor about α^{-1/2}. Moreover, the subsequent application estimates Σ_{n∈M} d(n)n^{-3/4}, M≍α/√log_2α (logα)^{λ−1−λlogλ}, and the final α^{1/4}(log)^{...} bound are all computed with n≈α. If M is instead chosen consistently with Theorem 6, i.e. n≈α², those estimates read differently (Σ a_n≈α^{1/2} times logs), and the stated lower bound is not obtained. The same mismatch appears in §4.2, §4.3, and the mean-square applications via the same M construction. Hence the central claim of the paper, as written, is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12899,"tokens_out":6218,"duration_ms":52942,"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":[{"comment":"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.","section":"Section 4.1, applied uniformly in Sections 4.2-4.5"},{"comment":"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.","section":"Section 3, proof of Theorem 6"}],"minor_comments":[{"comment":"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}\".","section":"Theorem 6 statement"},{"comment":"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.","section":"Section 4.4, equation (6)"},{"comment":"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.","section":"Section 2.2, footnote 2"}],"recommendation":"reject","confidential_remarks":"The two major comments above are independent and together invalidate the central theorem and all applications as written. The scaling mismatch is not a minor presentational issue; it changes the order of magnitude of the main term. I would not recommend further review until the M-construction is reconciled with the spectral window in Theorem 6."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper claims to improve Soundararajan's Omega bounds for the divisor and circle problems by replacing (log_3 x)^{-5/8} with (log_3 x)^{-3/8}, with analogous gains for Piltz divisors and mean-square zeta/L errors. The intended results are new and the general resonance lemma (Theorem 6) is a reasonable tool. The exposition is clear and prior work is properly credited.\n\nBut as written, the proof has a load-bearing flaw. Theorem 6's main term is (π/4e) Σ_{λ_n ∈ M} a_n, obtained using r(λ_n) ≥ e^{-1} and w_α(λ_n) ≥ απ. The second inequality is false on the stated range: w_α(λ) = π max(0, 2α − |λ − 2α|), so for 0 < λ ≤ 2α it equals πλ. You only get w_α(λ_n) ≥ απ if λ_n ≥ α, which the hypothesis λ_n ∈ [C1 α, 2α] does not guarantee unless C1 ≥ 1 (and the proof wants small C1).\n\nThe applications make this worse. In §4.1, M is defined as {λ_n : n ∈ [C1 α, 2α], ...} with λ_n = 4π√n. So when n is of size α, the frequencies λ_n are of size √α, not α. That contradicts the spectral condition in Theorem 6. All subsequent estimates (M count, Σ_{M} d(n)n^{-3/4}, the α^{1/4} main term) are computed with n ≈ α. If you instead choose M consistently with Theorem 6, i.e., n ≈ α², those estimates change and the stated bound does not follow. The same mismatch appears in §4.2 and §4.3, and the mean-square sections inherit it.\n\nSection 4.4 also says \"exact calculation ... can be carried out\" and then does not give it; that is a non-trivial gap.\n\nNone of this is cosmetic. The main term is where these errors land. I do not think the current version can be accepted. That said, the intended improvement is genuine, and the general framework is not obviously wrong—it needs a rescaling of the resonator set or a different spectral window. A carefully revised version that fixes the λ_n vs n mismatch and supplies the missing zeta calculation could be a real contribution.\n\nIf I were the editor, I would still send it to a referee rather than desk-reject, because the questions are important and the flaw is specific enough to be repairable. For now, though, I would not cite it.\n\nBest,\n[You]","headline":"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.","tokens_in":13453,"tokens_out":3105,"would_cite":false,"duration_ms":25249,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N37","11M06","11P21"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sharper omega bounds for divisor and circle problems follow from a new resonance theorem.","keywords":["omega results","resonance method","Dirichlet divisor problem","Gauss circle problem","Piltz divisor problem","mean square of Riemann zeta function","exponential sums","lower bounds"],"falsifier":"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.","tokens_in":12325,"feed_emoji":"🔢","tokens_out":13626,"duration_ms":114962,"temperature":0.7,"pith_summary":"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.","feed_headline":"Divisor and circle error terms get sharper omega bounds","feed_subtitle":"The resonance method improves the triple-log exponent from -5/8 to -3/8 in five classical error terms.","key_machinery":"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 $\\frac{\\pi}{4e}\\sum_{\\lambda_n\\in M}a_n$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies the baseline Omega-bounds for the divisor and circle problems and the construction that the paper adapts.","marker":"[16]"},{"why":"It provides the specific choice of resonator used in the applications.","marker":"[2]"},{"why":"It supplies the construction of M from integers with a prescribed number of prime factors.","marker":"[17]"},{"why":"It provides the Sathe-type estimate for the count of integers with a given number of prime factors, which sizes M.","marker":"[15]"},{"why":"It supplies the convolution formula for the resonator integral used in Lemma 1.","marker":"[13]"},{"why":"It provides the reduction and the previous bound for the mean-square error of the Riemann zeta function that Theorem 4 improves.","marker":"[11]"},{"why":"It provides the analogous reduction and previous bound for the mean-square error of Dirichlet L-functions used in Theorem 5.","marker":"[12]"},{"why":"It gives the estimate for integers with a prescribed number of prime factors in the residue class 1 mod 4 used in the circle-problem resonator.","marker":"[18]"},{"why":"It supplies the Voronoi summation formula that turns Delta(x^2) into the exponential sum the resonance method attacks.","marker":"[19]"}],"fun_headline_variants":["Resonance method tightens omega bounds for divisor and circle","Sharper omega for divisor and circle via resonance","Resonance method shaves triple-log exponent for five error terms","Resonance method yields sharper omega for five error terms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Resonance method tightens omega bounds for divisor and circle","Sharper omega for divisor and circle via resonance","Resonance method shaves triple-log exponent for five error terms","Resonance method yields sharper omega for five error terms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000917,"raw_usage":{"total_tokens":3916,"prompt_tokens":906,"completion_tokens":3010,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":2942}},"tokens_in":522,"tokens_out":3010,"duration_ms":19669,"temperature":1.0,"reasoning_tokens":2942,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:52:22.741198+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Soundararajan","cited_arxiv_id":null,"evidence_quote":"It supplies the baseline Omega-bounds for the divisor and circle problems and the construction that the paper adapts."},{"cited_title":"Aistleitner and K","cited_arxiv_id":null,"evidence_quote":"It provides the specific choice of resonator used in the applications."},{"cited_title":"Soundararajan","cited_arxiv_id":null,"evidence_quote":"It supplies the construction of M from integers with a prescribed number of prime factors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the Sathe-type estimate for the count of integers with a given number of prime factors, which sizes M."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the convolution formula for the resonator integral used in Lemma 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the reduction and the previous bound for the mean-square error of the Riemann zeta function that Theorem 4 improves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the analogous reduction and previous bound for the mean-square error of Dirichlet L-functions used in Theorem 5."},{"cited_title":"Tenenbaum Introduction to analytic and probabilistic number theory","cited_arxiv_id":null,"evidence_quote":"It gives the estimate for integers with a prescribed number of prime factors in the residue class 1 mod 4 used in the circle-problem resonator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the Voronoi summation formula that turns Delta(x^2) into the exponential sum the resonance method attacks."}],"review_version":1}