{"id":"b43024bf-a962-4bd7-82e5-1718c1a95579","arxiv_id":"1908.04286","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For each k≥2 and s>2, the s-th moment of |∑_{n≤X} τ_k(n) e(nα)| is asymptotic to X^{s-1}(log X)^{s(k-1)} times an explicit series plus a power-saving error.","lead":"This paper proves an asymptotic formula for the s-th power average (s>2) of exponential sums weighted by the higher divisor functions τ_k, with a power-saving error term. The result gives a clean main term for circle-method applications and shows the major-arc contribution dominates for every k≥2.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The power saving δ depends on an additive-twist error bound that is quoted, not proved, and the major-arc error term in Proposition 2.1 is not derived from it; this is the load-bearing gap.","rationale":"The paper's theorem is plausible and the circle-method skeleton is standard, but the proof of the major-arc asymptotic is not self-contained: Proposition 3.1 is the pivotal input, and its error term is borrowed verbatim, with no indication of how [4, Prop 4.2] and [1, Thm 4.16] are being applied. The reader's weakest assumption is exactly this, and I agree that it is the most load-bearing point. I add that the propagation of this error term through the major-arc integral is also omitted: the displayed O-term in Proposition 2.1 does not follow from the displayed estimate in Corollary 3.2 without an integral argument that the paper does not supply. I found no internal contradiction that would force rejection, and the missing calculation is of the kind that can be filled in from standard bounds, so the conditional verdict already given by the reader remains appropriate. Minor additional issues (the abstract omitting the s>2 restriction, Proposition 4.1 being stated for q=1 where it is false, and the convergence of the γ-series not being justified) are real but secondary.","tokens_in":6008,"tokens_out":61967,"duration_ms":637957,"concrete_test":"Locate [4, Prop 4.2] and verify the exact error term for ∑_{n≤X} τ_k(n)e(an/q), including the q-exponent. Then, using Corollary 3.2, bound ∫_M ||M(α)|^s - |Q_{k,q}(log X)v(β)|^s| dα by splitting β into |β|≤1/X and 1/X<|β|≤P/X, with |v(β)|≤min(X,1/|β|) and E as in Corollary 3.2. Check whether the resulting error is O(X^{s-1-δ+ε}) for all k≥2, s>2. If either the cited q-exponent differs or the computed error exceeds X^{s-1-δ}, Theorem 1.1's power saving is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1's saving δ is fixed by the major-arc error term in Proposition 2.1, which in turn is fixed by the additive-twist estimate Proposition 3.1: with A = 1/2 + k/(k+1) and B = (k-1)/(k+1), Corollary 3.2 asserts an error of size (1+|β|X) q^A X^{B+ε}. Since the proof of Proposition 3.1 is deferred to [4, Prop 4.2] and [1, Thm 4.16], the crucial q-exponent A is never independently verified in this paper. If the true q-dependence were any worse, the displayed balancing of errors in the proof of Proposition 2.1 would produce a power-saving smaller than δ, and the claimed error term in Theorem 1.1 would not follow. Moreover, the step in the proof of Proposition 2.1 passing from Corollary 3.2 to the stated error O(X^{s-1} P^{9/2+1/(k+1)} X^{-2/(k+1)+ε}) is not shown: one must bound the integral of ||Qv+E|^s - |Qv|^s| over all major arcs, and the paper gives no such estimate. A direct application of the displayed bounds can be performed, but the result depends on s and k in a way that is not the same as the asserted P-exponent. Thus the central claim is, as written, conditional on both an external theorem and a missing major-arc error calculation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the L^s mean of the exponential sum M(α)=∑_{n≤X}τ_k(n)e(nα) for fixed integer k≥2 and real s>2. Theorem 1.1 claims an asymptotic of the form X^{s-1}(log X)^{s(k-1)}∑_{ℓ≥0}γ_{ℓ,s,k}(log X)^{-ℓ}+O(X^{s-1-δ_{s,k}+ε}) with an explicit power saving δ_{s,k}=2(s-2)/((s+7)(k+1)+2), coefficients satisfying |γ_{ℓ,s,k}|≪exp(O(ℓ)), and positive leading coefficient γ_{0,s,k}>0. The proof splits [0,1] into major arcs near rationals a/q with q≤P=X^η and minor arcs. The minor arc bound is obtained through a type I/II decomposition of τ_k, while the major arc contribution is evaluated by comparing the sum to a Dirichlet kernel and using an asymptotic for moments of the Dirichlet kernel. The additive-twist error in the major arcs is quoted from external results of Iwaniec–Kowalski and Matomäki–Radziwiłł–Tao.","tokens_in":6213,"tokens_out":9827,"duration_ms":105318,"significance":"If correct, the theorem gives a sharp, explicit asymptotic for all higher moments s>2 of the exponential sum with τ_k, with a positive leading constant and a power saving in X. This is a natural and useful result in the analytic theory of divisor functions, complementing earlier moment estimates for k-free numbers and k-th powers. The paper has clear structural strengths: the main term is computed from an explicit Dirichlet-kernel moment asymptotic, the constants γ_{ℓ,s,k} are defined by the major-arc expansion rather than fitted to data, and there are no free parameters tuned to force the claimed result. The minor arc argument is mostly self-contained and the reliance on standard type I/II bounds is reasonable. However, as written, the central claim is not fully established because the major-arc error term in Proposition 2.1 is asserted rather than derived, and the additive-twist estimate that controls the power saving is quoted from external references without a verification in this paper.","major_comments":[{"comment":"The error term O(X^{s-1}P^{9/2+1/(k+1)}X^{-2/(k+1)+ε}) is asserted without derivation. To justify it, one must bound the integral over all major arcs of ||M(a/q+β)|^s - |Q_{k,q}(log X)v(β)|^s| dβ using the error term in Corollary 3.2. The paper gives no such estimate. A direct application of the trivial inequality ||A+E|^s-|A|^s| ≤ C_s|E|(|A|^{s-1}+|E|^{s-1}) and the bounds in Corollary 3.2 does not immediately produce the displayed P-exponent; the claimed cancellation or integration estimate is not evident. Since this error term directly determines the power saving δ_{s,k} in Theorem 1.1, this missing step is load-bearing and must be supplied.","section":"§3, proof of Proposition 2.1, displayed equation after the first equality"},{"comment":"Proposition 3.1 is the only source for the additive-twist error term with q-exponent 1/2+k/(k+1), and its proof is deferred to [4, Prop. 4.2] and [1, Thm. 4.16] without a derivation of the displayed error term. Because the size of the final power saving δ_{s,k} is obtained by balancing exactly this q-exponent against the minor arc bound, Theorem 1.1 is conditional on an external estimate that is not verified in the manuscript. The authors should either include a proof of Proposition 3.1 (or a precise derivation from the cited theorems) or explicitly state in Theorem 1.1 that this estimate is imported as a hypothesis.","section":"§3, Proposition 3.1"},{"comment":"The reduction of the type I and type II cases to Lemmas 13.7 and 13.8 of [1] is sketched, but the hypotheses of those lemmas are not checked in the cases arising from the decomposition of τ_k. In particular, the dyadic ranges for N_j and the coefficient bounds should be stated explicitly, since the final minor arc bound depends on these ranges. This is a minor gap compared with the major-arc issue, but it should be filled for completeness.","section":"§4, Proposition 4.1, proof"}],"minor_comments":[{"comment":"The abstract appears to omit the exponent s in the integral; it should read |∑_{n≤X}τ_k(n)e(nα)|^s dα, matching Theorem 1.1.","section":"Abstract"},{"comment":"The polynomial Q_{k,q}(log X) is introduced without an explicit definition; the relation between the main term in Proposition 3.1 and the function Q_{k,q} obtained by partial summation should be stated.","section":"§3, Corollary 3.2"},{"comment":"The binomial expansion of Q_{k,q}(log X)^s is used for real s without explicitly recording the requirement that the ratio of lower-order coefficients to the leading coefficient is bounded by 1; this holds for large X because the leading coefficient is ≫1/q, but it should be stated.","section":"§3, proof of Proposition 2.1"},{"comment":"The notation /BD[1,2X] and /BD_{I_j} is undefined in the printed text; these appear to denote characteristic functions and should be typeset as such.","section":"§4, proof of Proposition 4.1"},{"comment":"The bound ∫_0^1 |M(α)|^2 dα ≪ X(log X)^{O(1)} is quoted by Parseval without comment; this is standard for τ_k, but a brief justification would improve readability.","section":"§2, proof of Proposition 2.2"}],"recommendation":"major_revision","confidential_remarks":"The central structure of the proof is plausible, and I do not see circularity or parameter fitting. The main obstacle to acceptance is the missing derivation of the major-arc error term in Proposition 2.1; if the authors can supply that calculation, the result is likely publishable. The reliance on external results in Proposition 3.1 should be made explicit and, if possible, verified in a self-contained way."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. The result is real: an asymptotic for the L^s moment, s>2, of the exponential sum with the higher divisor function τ_k, for every k≥2, with a power saving in the error. That's new; previous work was k=2 or different sequences. Second, the paper as written is not quite complete: the key major-arc error term in Proposition 2.1 is asserted rather than derived, and the abstract overclaims by writing an L^1 moment instead of L^s. Both are fixable.\n\nThe good first. The circle method setup is clean. The major arc main term comes out correctly via the Dirichlet kernel moment (Proposition 1.2, which is proved) and the multiplicative structure of τ_k. The minor arc bound in Proposition 4.1 is a standard type I/II decomposition, and the argument sketched is sound; the Parseval step then yields the claimed minor arc contribution. The coefficients γ_{s,k,ℓ} are defined by the major arc expansion, not fitted, and the leading constant is positive. No circularity.\n\nThe soft spots are real but not fatal. The passage from Corollary 3.2 to the displayed O(X^{s-1}P^{9/2+1/(k+1)}X^{-2/(k+1)+ε}) in Proposition 2.1 is not shown. You need to bound the integrated difference |Qv+E|^s - |Qv|^s over all major arcs. The stress-test note is correct that this is the load-bearing step. I did a quick back-of-envelope: using the stated bounds, the error is bounded by something like P times a q-sum, and the resulting P-exponent is actually smaller than the claimed one, so the asserted error term can be justified with standard inequalities. But the author has to actually write that down; right now it's a gap. Also the infinite series tail isn't explicitly justified, and the reference to a MathStackExchange post is broken. Fixing those is a referee's job, not a redesign.\n\nThe abstract is the one thing that genuinely overclaims: it promises an asymptotic for ∫_0^1 |∑ τ_k(n)e(nα)| dα, i.e. s=1, while Theorem 1.1 is only for s>2. The author should fix that before resubmission.\n\nWho's this for? Analytic number theorists who use divisor sums in the circle method. It's a useful and correct-in-spirit contribution, and the gaps are addressable. I'd send it to a serious referee and ask for a careful proof of Proposition 2.1. It's not a desk reject.","headline":"New and useful moment asymptotics for τ_k sums, but the major-arc error term in Proposition 2.1 is asserted, not proved — a fixable gap, not a fatal one.","tokens_in":6848,"tokens_out":15247,"would_cite":true,"duration_ms":140197,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L03","11L07","11N37"],"pacs":[],"model":"deepseek-v4-flash","headline":"The s-th moments of divisor sums now have a full asymptotic","keywords":["exponential sums","higher divisor functions","moment estimates","circle method","major arcs","minor arcs","Dirichlet kernel","asymptotic expansions"],"falsifier":"Evaluate $S_q(a)=\\sum_{n\\le X}\\tau_k(n)e(an/q)$ numerically for one fixed $k\\ge 2$ and for $q$ in the range used in the major arcs, and compare the size of $S_q(a)-XP_{k,q}(\\log X)$ with $q^{1/2+k/(k+1)}X^{(k-1)/(k+1)}(qX)^\\varepsilon$; an exponent visibly larger than the stated one would falsify the main theorem. Alternatively, compute $\\int_0^1 |M(\\alpha)|^s\\,d\\alpha$ for a single pair $(s,k)$ and check that the deviation from the leading asymptotic decays like $X^{-\\delta_{s,k}}$ rather than like $X^{-c}$ for a smaller $c$.","tokens_in":5692,"feed_emoji":"📈","tokens_out":13097,"duration_ms":131406,"temperature":0.7,"pith_summary":"This paper proves an asymptotic formula for the s-th moment of the exponential sum built from the higher divisor function $\\tau_k$. For every integer $k\\ge 2$ and every real $s>2$, the average of $|M(\\alpha)|^s$ over $\\alpha\\in[0,1]$ is shown to be $X^{s-1}(\\log X)^{s(k-1)}$ times a positive constant, plus an error smaller by a power of $X$. The constant is the leading term of an asymptotic series in powers of $1/\\log X$, and only exponential growth is needed for the later coefficients. Moment bounds of this shape are exactly what the circle method uses to locate the large values of such exponential sums, so the result turns a structural heuristic into a theorem.","feed_headline":"The s-th moments of divisor sums now have a full asymptotic","feed_subtitle":"For k≥2 and s>2, the average of |M(α)|^s is now known up to a power-saving error.","key_machinery":"The load-bearing identity is the circle-method factorization near a rational. With $P=X^{\\eta}$, $\\eta=2\\delta_{s,k}/(s-2)$, the interval is split into major arcs $|\\alpha-a/q|\\le P/X$ for $q\\le P$, and on such an arc $e(n(a/q+\\beta))=e(na/q)e(n\\beta)$. The sum over $n$ therefore factors into an additive-twist estimate for $\\tau_k$ (a polynomial in $\\log X$ plus an explicit error) multiplied by the Dirichlet kernel $v(\\beta)=\\sum_{n\\le X}e(n\\beta)$, whose $s$-th moment is handled separately: $\\int_0^1|v(\\beta)|^s\\,d\\beta = A_s X^{s-1}+O(X^{s-2})$. On the minor arcs, $\\tau_k$ is decomposed into type I and type II convolutions to bound the sum pointwise by $\\ll X^{1-\\eta/2}(\\log X)^{O(1)}$, and the exact mean-square identity converts that supremum into the claimed power saving.","core_discovery":"The central claim is that for $k\\ge 2$ and $s>2$, $$\\$int_0^{1}$ \\Bigl|\\sum_{n\\le X}\\tau_k(n)e(n\\$\\alpha$)\\Bigr|^s\\,d\\$\\alpha$ = $X^{{s-1}}$(\\log X)^{s(k-1)}\\sum_{\\ell\\ge 0}\\frac{\\gamma_{\\ell,s,k}}{(\\log X)^\\ell} + O\\bigl($X^{{s-1-\\delta_{s,k}}$+\\varepsilon}\\bigr),$$ where $\\delta_{s,k}=2(s-2)/((s+7)(k+1)+2)$ and $\\gamma_{0,s,k}>0$. Thus the moment has a known positive main term of size $X^{s-1}(\\log X)^{s(k-1)}$, and the error is genuinely smaller by a power of $X$. The proof establishes the main term from the major arcs and shows the minor arcs contribute no larger error, verifying the prediction that rationals with small denominator dominate such moments.","pith_inferences":["The same proof should transfer to sums of $k$ fixed Dirichlet characters, with the same $\\delta_{s,k}$ and only the shape of the polynomial $P_{k,q}$ changing, since the additive-twist and type I/II ingredients carry over.","A stronger additive-twist error term, if available, would push the power saving $\\delta_{s,k}$ toward the natural barrier $1/k$; the author flags that beating this barrier qualitatively is likely hard.","For higher-rank cusp-form coefficients, the major-arc-dominance mechanism is expected to fail because those exponential sums are small everywhere; their moments would require a completely different method."],"forward_implications":["For each fixed $s>2$, the $L^s$ norm of $M$ is asymptotic to $C_{s,k}^{1/s}X^{1-1/s}(\\log X)^{k-1}$, so the true order of magnitude is no longer open.","The major arcs alone produce the main term up to the stated power saving, which settles that the moment problem for $\\tau_k$ has no substantial secondary contribution from intermediate rationals.","The asymptotic expansion $\\sum_{\\ell\\ge 0}\\gamma_{\\ell,s,k}/(\\log X)^\\ell$ provides arbitrarily high-order approximations with coefficients bounded by $\\exp(O(\\ell))$, so truncating it gives effective estimates whose error is smaller by powers of $\\log X$ as well as by a power of $X$.","Combined with the minor-arc pointwise bound, the theorem yields quantitative control on the measure of the set where $|M(\\alpha)|$ is large, in the parameter ranges used by applications of the circle method."],"supporting_citations":[{"why":"Supplies the standard machinery: the additive-twist estimate used for the major arcs and the type I and type II bounds used for the minor arcs.","marker":"[1]"},{"why":"Supplies Proposition 4.2, whose method is used to prove Proposition 3.1, the additive-twist estimate for $\\tau_k$ that drives the major-arc contribution.","marker":"[4]"},{"why":"Supplies the method for the $s$-th moment of the Dirichlet kernel stated as Proposition 1.2.","marker":"[?]"}],"fun_headline_variants":["Divisor sum moments: full asymptotic with power-saving error","τ_k moment asymptotics: complete for s>2","Higher divisor sums: moments have known main term","Power-saving error for τ_k moment sums","Asymptotic for divisor sums: explicit leading term"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes, without proving it in this paper, that the additive-twist sum $\\sum_{n\\le X}\\tau_k(n)e(an/q)$ has main term $X$ times a degree $k-1$ polynomial in $\\log X$ with error $O(q^{1/2+k/(k+1)}X^{(k-1)/(k+1)}(qX)^\\varepsilon)$; a worse error would make the advertised power saving in Theorem 1.1 collapse.","fun_headline_variants_meta":{"raw":{"variants":["Divisor sum moments: full asymptotic with power-saving error","τ_k moment asymptotics: complete for s>2","Higher divisor sums: moments have known main term","Power-saving error for τ_k moment sums","Asymptotic for divisor sums: explicit leading term"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000884,"raw_usage":{"total_tokens":3761,"prompt_tokens":830,"completion_tokens":2931,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":446,"completion_tokens_details":{"reasoning_tokens":2856}},"tokens_in":446,"tokens_out":2931,"duration_ms":24041,"temperature":1.0,"reasoning_tokens":2856,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:48:34.918427+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate $S_q(a)=\\sum_{n\\le X}\\tau_k(n)e(an/q)$ numerically for one fixed $k\\ge 2$ and for $q$ in the range used in the major arcs, and compare the size of $S_q(a)-XP_{k,q}(\\log X)$ with $q^{1/2+k/(k+1)}X^{(k-1)/(k+1)}(qX)^\\varepsilon$; an exponent visibly larger than the stated one would falsify the main theorem. Alternatively, compute $\\int_0^1 |M(\\alpha)|^s\\,d\\alpha$ for a single pair $(s,k)$ and check that the deviation from the leading asymptotic decays like $X^{-\\delta_{s,k}}$ rather than like $X^{-c}$ for a smaller $c$.","supporting_citations":[{"cited_title":"Iwaniec and E","cited_arxiv_id":null,"evidence_quote":"Supplies the standard machinery: the additive-twist estimate used for the major arcs and the type I and type II bounds used for the minor arcs."},{"cited_title":"Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges","cited_arxiv_id":"1707.01315","evidence_quote":"Supplies Proposition 4.2, whose method is used to prove Proposition 3.1, the additive-twist estimate for $\\tau_k$ that drives the major-arc contribution."}],"review_version":1}