{"id":"a1e33ade-e6bf-48af-8134-6eaf0706a514","arxiv_id":"2506.09334","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For every k>2, unconditionally, the average of |∑_{n≤x} n^{-it}|^{2k} over t∈[0,T] is ≫_k x^k (log L)^{(k-1)^2}, where L = min{x, T/x}.","lead":"The paper proves unconditional lower bounds for the 2k-th moments of zeta sums, the sums over n≤x of n^{-it}. The lower bound matches the best known conditional upper bounds and fixes the order of magnitude of these high moments for every k>2.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1's proof is incomplete: the Error in the integral-to-expectation reduction is never estimated, and the decisive lower bound is delegated to [14], so Theorem 1.1 is not self-contained.","rationale":"I read the proof as a standard Hölder argument with a proxy object R(t). The upper-bound side (Proposition 2) is sketched plausibly, although Lemma 3.2 also uses the same unproved mean-value equality and the partition into the sets T(n_1,...,n_M) is written very tersely. The load-bearing step for a lower bound is Proposition 1: the claimed reduction to a Steinhaus expectation is the only place where the main term x(log y)^{k^2-1} enters. If Error is not bounded by o(x(log y)^{k^2-1}), the implication 'it is sufficient to show the expectation lower bound' does not follow. Lemmas 2.1 and 2.2 appear intended to provide exactly this input, but they are stated without proof and are not cited in Proposition 1. The delegation to 'the proof of Proposition 3.1 in [14]' may be acceptable in a research announcement, but it leaves the central claim unverified. Since this matches the reader's weakest assumption and does not identify a new independent flaw, I would keep the conditional verdict.","tokens_in":4699,"tokens_out":14898,"duration_ms":162234,"concrete_test":"Estimate the off-diagonal contribution in Proposition 1. Expand |Σ_{n≤x} n^{-it}|^2 R(t) as a Dirichlet polynomial, and use Montgomery-Vaughan's mean value theorem (or the mean-value argument underlying [14, Prop. 3.1]) to bound Error = (1/T)∫_0^T |Σ_{n≤x} n^{-it}|^2 R(t) dt − E|Σ_{n≤x} f(n)|^2 R(f). Check whether, for C0 sufficiently large and y=x^{1/C0}, |Error| ≤ (1/2) x (log y)^{k^2-1}. If yes, Proposition 1 is established; if not, the proof has a material gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 1 is the only source of the lower bound in Theorem 1.1, yet its proof consists of two assertions. First, it claims (1/T)∫_0^T |Σ_{n≤x} n^{-it}|^2 R(t) dt = E|Σ_{n≤x} f(n)|^2 R(f) + Error, with no estimate for Error; if Error has size comparable to the main term or is negative, the lower bound is void. Second, it asserts the expectation lower bound follows immediately from the proof of Proposition 3.1 in [14]; but Lemmas 2.1 and 2.2, which are the stated random-model inputs, are not proved and are never invoked in the proof. The adaptation of Szabó's argument to the Steinhaus zeta-sum setting (with y=x^{1/C0}, the truncation parameters J_m, and the outer ℓ-sum) is not shown. Since Hölder plus Proposition 2 cannot compensate for a missing lower bound, the central claim is currently unsupported at its decisive step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims an unconditional lower bound for the high moments of zeta sums: for large T and any k > 2, (1/T) ∫_0^T |∑_{n≤x} n^{-it}|^{2k} dt ≫_k x^k (log L)^{(k-1)^2}, where L = L(x,T) = min{x, T/x}. The method is to apply Hölder's inequality with a carefully chosen proxy object R(t), prove a lower bound for the weighted mean square of the zeta sum against R(t) (Proposition 1), and prove an upper bound for the integral of R(t)^{k/(k-1)} (Proposition 2). Both propositions are supposed to follow by transferring known estimates for the Steinhaus random multiplicative function from Szabó's work on character sums to zeta sums, but in the submitted manuscript the transfer is asserted rather than proved.","tokens_in":4958,"tokens_out":4042,"duration_ms":45147,"significance":"If the proof can be completed, Theorem 1.1 is a significant result: it would remove the Riemann hypothesis from Gao's conditional upper bound and establish the conjectured order of magnitude of high moments of zeta sums for k > 2. The high-level strategy is elegant and the paper is concise, with the proxy construction and the Hölder argument clearly laid out. However, the decisive lower-bound step is not actually proved in the manuscript: Proposition 1 rests on an unquantified error term and on an undeveloped appeal to the proof of Proposition 3.1 in [14]. The same applies to the random-model input in Lemma 3.2. The manuscript therefore currently functions as a research announcement rather than a complete proof.","major_comments":[{"comment":"The proof of Proposition 1 begins with the relation (1/T)∫_0^T |∑_{n≤x} n^{-it}|^2 R(t) dt = E |∑_{n≤x} f(n)|^2 R(f) + Error, but no estimate for Error is given. Since Proposition 1 is the only source of the lower bound in Theorem 1.1, a non-negligible or negative Error would invalidate the claimed lower bound. The author must either prove that Error = o(x (log y)^{k^2-1}) with explicit hypotheses, or replace this assertion by a fully stated and proved lemma. As written, the main theorem is unsupported at its decisive step.","section":"Section 3, Proposition 1"},{"comment":"The sentence 'This follows immediately from the proof of Proposition 3.1 in [14]' is not a proof in the present manuscript. Lemmas 2.1 and 2.2, which are the stated random-model inputs, are neither proved nor invoked in the displayed argument, and the adaptation of Szabó's character-sum argument to the Steinhaus setting with the specific choices y = x^{1/C0}, the truncation parameters J_m, and the outer ℓ-sum is not demonstrated. The authors need to include the derivation or explicitly state the needed result as a theorem with all hypotheses and a proof.","section":"Section 3, Proposition 1"},{"comment":"Lemma 3.2 is proved by first asserting an integral-to-expectation inequality and then saying that 'the lemma follows immediately from the proof of Proposition 5.1 in [14].' No error term is bounded in the passage from the zeta-sum integral to the Steinhaus expectation, and the applicability of Szabó's argument to the present R_{m,ℓ} and U_{m,ℓ} is not verified. Since Lemma 3.2 is the key input in Proposition 2, the upper bound needed for the Hölder factor is not fully justified. This is a load-bearing gap, not a presentation issue.","section":"Section 3, Lemma 3.2"}],"minor_comments":[{"comment":"The Introduction contains typos: 'Stenhaus' should be 'Steinhaus' and 'Radmacher' should be 'Rademacher'.","section":"Section 1"},{"comment":"The displayed definition of E_{m,ℓ}(f) appears garbled: the right-hand side is written as exp(...) - R_{m,ℓ}(f) = sum over u,v, with R_{m,ℓ}(f) already used earlier as a truncated exponential. Please rewrite the definition so that the remainder term is unambiguous.","section":"Section 2, Lemma 2.2"},{"comment":"The second case in the definition of U_{m,ℓ} has a typographical formatting issue; the inequalities 'J_m/(100k) ≤ A_m ≤ 100 k J_m' and '100 k J_m ≤ A_m' should be displayed clearly so that the piecewise definition is readable.","section":"Section 3, definition of U_{m,ℓ}"},{"comment":"The displayed Hölder inequality before Proposition 1 is not numbered; numbering it would make the cross-reference in Proposition 1's proof cleaner.","section":"Section 3, displayed Hölder inequality"},{"comment":"Reference [13] is missing journal and volume data, and the theorem quoted from Gao is labeled simply 'Theorem.' before the introduction; please number it for clarity.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claim is plausible and the overall strategy is promising, but the proof as submitted is incomplete at the key transfer step: Proposition 1 is essentially two assertions with no estimates and no proof of the lower bound for the Steinhaus expectation. This is not a matter of polish; the main theorem currently depends on an unproved black box. I would not send the paper to production in this form. Major revision, with the missing arguments supplied, seems appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper states the expected unconditional lower bound for high moments of zeta sums, matching Gao's conditional upper bound and generalizing Szabó's character-sum lower bound. The result is very likely true. But the proof as written is not self-contained at exactly the load-bearing steps, so treat Theorem 1.1 as a plausible theorem with a sketched proof rather than a complete proof.\n\nWhat is genuinely new: Theorem 1.1, the first unconditional lower bound for these moments. The strategy is standard after Szabó: Hölder with a proxy R(t) built from short Dirichlet polynomials, then a lower bound for |S|^2 R integrated, and an upper bound for R^{k/(k-1)}. Proposition 2 and Lemma 3.1 are actually proved in the text, and the argument there is clear. The definitions (subdivision ratio 20, truncation parameters J_m, the ℓ-sum) all follow Szabó's paper [14] closely, which is fine because that paper is the right template.\n\nWhere it gets soft: Proposition 1, the entire source of the lower bound, has a two-line proof. First it asserts an integral-to-expectation relation with an Error term that is never even written out, let alone bounded. For the lower bound to survive, this Error needs to be o(x (log y)^{k^2-1}). Second, after that reduction it says 'this follows immediately from the proof of Proposition 3.1 in [14]'. But the random model here is Steinhaus zeta sums, not character sums, and the parameters y = x^{1/C0}, the J_m, and the outer ℓ-sum need to be checked. The same pattern appears in Lemma 3.2, which transfers to 'Proposition 5.1 in [14]' with no details. The two preliminary lemmas (2.1, 2.2) are stated but never proved and never actually invoked in the text. So the reader who wants to verify the paper has to do the adaptation themselves.\n\nI don't think this is a sign the result is wrong. The structure is too deliberate for that, and the missing steps are exactly the kind that Szabó's methods should supply. But it's a real gap, not a cosmetic one. A referee would need to see the Error bound and the adapted arguments before the proof is complete.\n\nRecommendation: send it to peer review, but with an explicit request that the authors fill in the transfers from [14] and bound the error in Proposition 1. If they can do that, the paper is a solid contribution. As it stands, it's a valuable announcement with a sketch.","headline":"New unconditional lower bound for high moments of zeta sums, matching Gao's conditional upper bound, but the proof delegates the load-bearing steps to Szabó's character-sum paper and never estimates the transfer error.","tokens_in":5411,"tokens_out":2925,"would_cite":true,"duration_ms":28590,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L40","11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves unconditional lower bounds for high moments of zeta sums, matching the upper bounds that were previously known only conditionally on the Riemann hypothesis.","keywords":["zeta sums","high moments","unconditional lower bounds","random multiplicative functions","Dirichlet polynomials","Riemann hypothesis","character sums","Hölder's inequality"],"falsifier":"Take a sequence of pairs $(T, x)$ with $x = T^{1/2}/(\\log T)^{10}$ (so $L \\approx x$ for large $T$) and compute the normalized moment $\\frac{1}{T}\\int_0^T |\\sum_{n\\le x} n^{-it}|^{2k}\\,dt \\,/\\, [x^k(\\log L)^{(k-1)^2}]$ for $k=3$. The theorem asserts this ratio has a positive liminf as $T\\to\\infty$; if the ratio tends to zero along any sequence, the central claim is false.","tokens_in":4520,"feed_emoji":"📈","tokens_out":11526,"duration_ms":103343,"temperature":0.7,"pith_summary":"The paper proves that the high moments of the zeta sum $\\sum_{n\\le x} n^{-it}$, averaged over $t$ in $[0,T]$, grow at least like $x^k (\\log \\min\\{x,T/x\\})^{(k-1)^2}$ for every $k>2$, with no unproved hypothesis. This is the same order of magnitude as the previously known upper bound conditional on the Riemann hypothesis, and it matches what the random multiplicative function model predicts. The proof works by bounding a carefully chosen proxy object $R(t)$ from above in the mean and from below when weighted by the zeta sum, then applying H\\\"older's inequality. The lower-bound part is imported from the paper's companion treatment of character sums, after transferring the integral over $t$ to an expectation over a random multiplicative function.","feed_headline":"Zeta sum high moments get unconditional lower bounds","feed_subtitle":"Order of magnitude now fixed from below, matching the conditional upper bound known under RH.","key_machinery":"The proxy object is $R(t) = \\sum_{|\\ell|\\le (\\log y)/2} \\prod_{m=1}^M R_{m,\\ell}(t)$, where each $R_{m,\\ell}(t)$ is the square of a truncated exponential in $\\Re D_{m,\\ell}(t)$, and $D_{m,\\ell}(t)$ is a short Dirichlet polynomial over primes in the dyadic block $(y_{m-1}, y_m]$. The parameter $y = x^{1/C_0}$ splits the primes into blocks, and the truncation parameters $J_m$ are chosen so that $\\prod_m y_m^{10^4 k J_m} < x$. H\\\"older's inequality with exponents $k$ and $k/(k-1)$ couples this proxy to the zeta sum: once the mean of $R(t)^{k/(k-1)}$ is shown to be $\\ll_k (\\log y)^{k^2+1}$ and the weighted mean of $|\\sum_{n\\le x} n^{-it}|^2 R(t)$ is shown to be $\\gg_k x (\\log y)^{k^2-1}$, the theorem follows.","core_discovery":"The central claim is that, for large $T$ and every $k>2$, the average $\\frac{1}{T}\\int_0^T |\\sum_{n\\le x} n^{-it}|^{2k}\\,dt$ is at least a positive constant (depending only on $k$) times $x^k (\\log L)^{(k-1)^2}$, where $L = \\min\\{x, T/x\\}$. This gives the same order of magnitude as the upper bound previously obtained under the Riemann hypothesis, so the lower bound is unconditional. The exponent $(k-1)^2$ is the novel part: it comes from the product over dyadic blocks of primes in a truncated-exponential proxy, and matches the random multiplicative model's high-moment prediction. The proof reduces the theorem to two estimates on the proxy $R(t)$: an upper bound for its $(k/(k-1))$-th moment and a lower bound for its weighted average against the zeta sum squared.","pith_inferences":["An explicit bound on the transfer error in Proposition 1 would make the proof self-contained and would likely yield effective implied constants in Theorem 1.1.","The same proxy-object construction should extend to shifted moments of zeta sums and to moments of derivatives of zeta functions, where currently only conditional upper bounds are known.","If this lower bound is sharp, then the high moments of zeta sums are now understood to the order of magnitude, leaving only the explicit constant open."],"forward_implications":["The high moments of zeta sums now have the same unconditional order of magnitude as the upper bound known under the Riemann hypothesis.","The lower bound holds for the full range of $x$ relative to $T$, with the logarithmic factor switching from $\\log x$ to $\\log(T/x)$ according to $L = \\min\\{x, T/x\\}$.","Together with the conditional upper bound, this pins the order of magnitude of the $2k$-th moment as $x^k(\\log L)^{(k-1)^2}$ for every $k>2$, assuming the Riemann hypothesis only for the upper side.","The H\\\"older-coupling method shows that the exponent $(k-1)^2$ arises from the product over dyadic prime blocks in the proxy, matching the random multiplicative model's prediction."],"supporting_citations":[{"why":"Supplies the character-sum lower-bound proof that the paper imports for Propositions 1 and 2.","marker":"[14]"},{"why":"Establishes the high-moment order for random multiplicative functions, the model whose exponent $(k-1)^2$ the zeta sums are shown to match.","marker":"[8]"},{"why":"Gives the conditional upper bound under the Riemann hypothesis that Theorem 1.1 matches unconditionally.","marker":"[5]"}],"fun_headline_variants":["Zeta sums: unconditional lower bound on high moments","High moments of zeta sums lower bounded without RH","Zeta sum high-moment order fixed unconditionally","Zeta sums: lower bounds for high moments, no RH","Unconditional result: zeta sum high moments"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that the difference between the $t$-average integral and the corresponding expectation over a random multiplicative function is negligibly small, and that the character-sum lower bound the paper cites transfers to this zeta-sum setting without a loss; neither the size of that difference nor the transfer is checked in the paper.","fun_headline_variants_meta":{"raw":{"variants":["Zeta sums: unconditional lower bound on high moments","High moments of zeta sums lower bounded without RH","Zeta sum high-moment order fixed unconditionally","Zeta sums: lower bounds for high moments, no RH","Unconditional result: zeta sum high moments"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000191,"raw_usage":{"total_tokens":1240,"prompt_tokens":738,"completion_tokens":502,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":354,"completion_tokens_details":{"reasoning_tokens":425}},"tokens_in":354,"tokens_out":502,"duration_ms":5432,"temperature":1.0,"reasoning_tokens":425,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:50:18.515691+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a sequence of pairs $(T, x)$ with $x = T^{1/2}/(\\log T)^{10}$ (so $L \\approx x$ for large $T$) and compute the normalized moment $\\frac{1}{T}\\int_0^T |\\sum_{n\\le x} n^{-it}|^{2k}\\,dt \\,/\\, [x^k(\\log L)^{(k-1)^2}]$ for $k=3$. The theorem asserts this ratio has a positive liminf as $T\\to\\infty$; if the ratio tends to zero along any sequence, the central claim is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the high-moment order for random multiplicative functions, the model whose exponent $(k-1)^2$ the zeta sums are shown to match."},{"cited_title":"Gao,Upper bounds for moments of zeta sums, J","cited_arxiv_id":null,"evidence_quote":"Gives the conditional upper bound under the Riemann hypothesis that Theorem 1.1 matches unconditionally."}],"review_version":1}