{"id":"f5208f7b-f878-4585-b741-b0ca333f214d","arxiv_id":"2509.09771","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New Omega results for maxima of Dirichlet polynomials with multiplicative coefficients in the range N ≤ sqrt(T) and in a transition regime, using resonance method and GCD sums.","lead":"This number theory paper proves new lower bounds for how large certain oscillating sums can grow when the coefficients are multiplicative and the sum length is up to the square root of the time range. The bounds are built with the resonance method and improve known results in a regime where the sum is relatively short.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's hypothesis F(c) is vacuous: the condition Re f(n) overline{f(m)} >= c forces f ≡ 1, so the advertised generality over multiplicative functions collapses.","rationale":"The Reader's verdict identified the I2 off-diagonal estimate in Theorem 1.1 as the weakest assumption, but that concern does not clearly land: in the I2 expansion, all products km and n are at most T, but n≤x=T/N, so when km≠n the closest possible ratio satisfies |log(km/n)| ≫ 1/x = N/T; the Fourier argument is therefore about N/log T, exactly as in I1, and the Gaussian decay makes the off-diagonal contribution negligible in the stated parameter range. The proof is terse and should be expanded, but it is not the central weakness. The truly load-bearing issue is that Theorem 1.2, the paper's main advertised improvement, is stated for a class F(c) that is actually the singleton {1}. The condition Re f(n) overline{f(m)} ≥ c for all m,n is incredibly strong: for a single prime p, it requires Re f(p)^k ≥ c for all k≥0, which forces f(p)=1. Hence every completely multiplicative f in F(c) is identically 1. This does not make the theorem false, because the lower bound may still hold for f=1, but it completely undermines the claimed generalization over multiplicative coefficients and the comparison with Xu–Yang's uniform result. A conditional acceptance is appropriate only if the authors reframe Theorem 1.2 around f≡1 or find a genuinely nonempty class of functions satisfying a repaired condition; as written, the central claim's advertised scope is vacuous.","tokens_in":7440,"tokens_out":28186,"duration_ms":336806,"concrete_test":"Settle the concern by checking whether any nonconstant f satisfies the F(c) condition. For a single prime p, set f(p)=e^{iθ}, θ≠0, and compute min_{1≤k≤K} Re(e^{ikθ}) for K large enough (e.g., K=⌈2π/|θ|⌉ or K equal to the order of e^{iθ} when θ is rational). This minimum will be < c for every c>0, confirming that F(c)={1}. Then, as a further verification, re-derive Theorem 1.2 for the surviving case f≡1 and compare the resulting range and exponent with Xu–Yang [14] to see whether the claimed improvement remains nontrivial without the misleading generality.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The main advertised improvement is Theorem 1.2, stated for every f in F(c). But F(c) contains no nonconstant function. Fix a prime p and write f(p)=e^{iθ}. Applying the defining condition with n=p^k and m=1 gives Re f(p^k) = Re(e^{ikθ}) >= c for every k ≥ 1. For θ ≠ 0, the positive powers of e^{iθ} cannot all lie in the half-plane Re z ≥ c > 0: if θ/π is irrational this follows from density of {kθ mod 2π}; if rational, taking k near half the order of e^{iθ} gives a value with real part < c. Hence θ = 0 for every prime p, and complete multiplicativity forces f ≡ 1. Thus the 'additional request' in Theorem 1.2 is not a mild restriction: it selects only the trivial function. The proof's step (3.8), which lower-bounds Re f(am) overline{f(bn)} by c, is therefore vacuous beyond the case f=1. The lower bound for the classical sum ∑_{n≤N} n^{it} may still be valid, but the central claim that this improves Xu–Yang's uniform result for all completely multiplicative f with |f|=1 is misleading, since the stated class is a singleton. This is more load-bearing than the I2 off-diagonal issue identified in the Reader's verdict: that issue appears repairable (the minimal phase gap in I2 is about N/log T, not O(1/T), because n≤x), whereas an empty hypothesis affects the interpretation of the main theorem directly.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies large values of Dirichlet polynomials ∑_{n≤N} f(n)n^{it} for completely multiplicative coefficients with |f(n)|=1. Theorem 1.1 claims a lower bound in the transition range log N = (log T log_2 T)^{1/2} τ with τ=(log_2 T)^{O(1)} for all such f. Theorem 1.2 claims a stronger lower bound, with exponent √2√(log(T/N) log_3(T/N)/log_2(T/N)), in the range exp((log T)^{1/2+δ}) ≤ N ≤ √T for all f in a class F(c) defined by Re f(n) overline{f(m)} ≥ c for all m,n. The proofs use the resonance method, estimates from Hough's work, and GCD-sum bounds of de la Bretèche–Tenenbaum.","tokens_in":7835,"tokens_out":15754,"duration_ms":173507,"significance":"If Theorem 1.2 held for a genuinely large class of multiplicative functions, it would improve on the uniform result of Xu and Yang. Theorem 1.1, if fully proved, would also be a useful extension in the transition regime. The paper correctly identifies the relevant external tools (Hough's resonator estimates and GCD-sum bounds) and the overall strategy is coherent. However, the central generalization claim of Theorem 1.2 is empty, because the class F(c) contains only the constant function f≡1 (and is empty for c>1). This directly undermines the advertised improvement over Xu–Yang. The paper also leaves the key off-diagonal estimate in Theorem 1.1 at the level of an assertion.","major_comments":[{"comment":"The class F(c) is essentially empty. For any prime p write f(p)=e^{iθ}. Since f is completely multiplicative, f(p^k)=f(p)^k. The condition Re f(n) overline{f(m)} ≥ c for all m,n, applied with m=1, gives Re f(p^k)=cos(kθ) ≥ c for every k≥1. If θ≠0 mod 2π, density (or rationality) gives some k with cos(kθ)<c. Hence θ=0 for every prime p, so f≡1. For c>1, F(c) is empty; for 0<c≤1, F(c)={1}. Thus Theorem 1.2 does not establish a result for a family of multiplicative functions; it reduces to f=1. The proof's step (3.8), which lower-bounds Re f(am) overline{f(bn)} by c, is vacuous beyond the constant function. This is a load-bearing error: the advertised improvement over Xu–Yang's uniform result is not realized.","section":"Section 3, definition of F(c) and Eq. (3.8)"},{"comment":"The sentence 'Similarly to our treatment for I1(R,T)' asserts the negligibility of the off-diagonal contribution to I2 without proof. In I1 the minimal logarithmic gap is c/x, giving a Gaussian argument cN/logT. In I2, when km≠n, the integers km and n differ by at least 1 and n≤x, so |log(km/n)| ≥ 1/(2x) and the Gaussian argument is at least N/(2 logT), which is large under the hypotheses. The step is therefore likely repairable. Nevertheless, the proof of Theorem 1.1 currently depends on an unverified claim at this point; a complete bound on the off-diagonal sum must be supplied.","section":"Section 2, after the expansion of I2"}],"minor_comments":[{"comment":"The citation 'similar to [?, Lemma 5]' is unresolved. Either replace it with a precise reference or remove it, since a proof is sketched.","section":"Section 3, proof of (3.3)"},{"comment":"There are numerous LaTeX corruption artifacts in the displayed equations (e.g., 'Nq 2 m/(m,n) n/(m,n)' and missing overlines in F(c)). These should be corrected in a revision.","section":"Throughout"},{"comment":"The final step 'Theorem 1.1 follows from [11, p.105–107] immediately' is very terse. The exact estimate needed for ∑_{k≤N} r(k) should be stated explicitly so the reader can verify the constant A(τ+τ′).","section":"Theorem 1.1"}],"recommendation":"reject","confidential_remarks":"The F(c) issue is not a cosmetic defect: it invalidates the main advertised theorem. The authors may be able to salvage Theorem 1.1, but Theorem 1.2 as stated is essentially a result for f≡1, which is a substantial change in scope rather than a local fix."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the punchline. The paper has a genuinely new lower bound for the classical zeta sum in the range exp((logT)^{1/2+ε}) ≤ N ≤ √T, and a transition-regime Omega result for arbitrary completely multiplicative f. But the advertised generality of Theorem 1.2 collapses: the class F(c) defined by Re f(n)f(m) ≥ c contains no nonconstant function. With m=1, n=p^k, the condition forces Re f(p)^k ≥ c for every k, which cannot hold for f(p)=e^{iθ} with θ≠0. Hence F(c) is either empty (c>1) or the singleton {1}. So 'for all f∈F(c)' is really 'for f=1'. That's not a mild 'additional request'—it defeats the stated improvement over Xu–Yang's uniform result.\n\nWhat's good: Theorem 1.1 is a reasonable new result, and the f=1 case of Theorem 1.2 goes beyond [14], whose parameter range excludes N near √T. The machinery—Hough's sums, the de la Bretèche–Tenenbaum GCD lemma, the resonance method—is appropriate, and the arithmetic is not circular. The self-citation is a baseline, not a crutch.\n\nSoft spots in proportion: the 'similarly' step in Theorem 1.1 is under-explained. The reader's worry about a O(1/T) phase gap is not right: the minimal nonzero gap is at least c/x, so the same Gaussian decay as in I1 applies. But the support property of r that makes it work should be stated. The placeholder '[?, Lemma 5]' is a real citation hole. And the F(c) vacuity is a load-bearing overstatement.\n\nNet: the core f=1 result is probably correct and new; the paper as written overclaims badly. A serious referee could get it into shape after major revision, with Theorem 1.2 restated honestly. I'd engage with a revised version, not this one.","headline":"Genuinely new zeta-sum bound, but Theorem 1.2's multiplicative-function class is a phantom: F(c) contains only f≡1.","tokens_in":8309,"tokens_out":7869,"would_cite":false,"duration_ms":87953,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L40","11N25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For multiplicative-coefficient Dirichlet polynomials, this paper proves large-value lower bounds across nearly the whole range exp((logT)^(1/2+ε)) ≤ N ≤ √T.","keywords":["Dirichlet polynomials","completely multiplicative functions","resonance method","GCD sums","Omega results","large values","zeta sums","mean values"],"falsifier":"Compute, for the r defined in Theorem 1.1 and logN = √(logT log_2 T), the off-diagonal sum ∑_{k≤N}∑_{m,n≤x, km≠n} f(km) overline{f(n)} r(m)r(n) Φ(T/logT log(km/n)) and compare it with ∑_{m≤x} r(m)^2. If this ratio does not tend to 0 as T→∞, the proof of Theorem 1.1 fails; a numerical check for moderate T would already indicate whether the claimed negligibility holds.","tokens_in":7360,"feed_emoji":"📈","tokens_out":5931,"duration_ms":61040,"temperature":0.7,"pith_summary":"The paper proves new Omega-type lower bounds for the maximum over 1 ≤ t ≤ T of Dirichlet polynomials ∑_{n≤N} f(n)n^{it}, where f is completely multiplicative and |f(n)|=1. In the main range, where N lies between exp((logT)^(1/2+ε)) and √T, and the coefficients satisfy a positive-correlation condition Re(f(n) overline{f(m)}) ≥ c for all m,n, the maximum is at least √N exp((√2+o(1))√(log(T/N) log_3(T/N)/log_2(T/N))). In the transition range logN = √(logT log_2 T) τ with τ=(log_2T)^{O(1)}, the bound √N exp((1+o(1))A(τ+τ')√(logT/log_2T)) holds for every completely multiplicative f with |f|=1. The proofs combine a resonance construction with estimates for GCD sums, and the key technical task is showing that off-diagonal terms in the second moment are negligible.","feed_headline":"Multiplicative Dirichlet sums beat √N by an exponential factor","feed_subtitle":"A resonance construction now covers N from exp((log T)^(1/2+ε)) up to √T, a much wider range than before.","key_machinery":"The resonator. For Theorem 1.2 it is constructed by taking a set M of ⌊T/N⌋ integers that maximizes the GCD sum (1/|M|)∑_{m,n∈M} √((m,n)/[m,n]), then thinning it to a well-separated subset M' by keeping the smallest element of each dyadic logarithmic bin and assigning weight r(m_j)=|M_j|^{1/2}. The Gaussian weight Φ(t)=e^{-t²/2} and its positive Fourier transform make the second moment a sum of terms Φ(T/logT · log(ma/nb)), whose rapid decay forces diagonal terms (am=bn) to dominate. For Theorem 1.1 the resonator uses a completely multiplicative r supported on primes in [λ², exp((logλ)²)], following Hough's construction, and the Gaussian factor bΦ(cN/logT) makes the off-diagonal contribution","core_discovery":"The central claim is that the resonance method can be made to work across the entire range N ≤ √T for Dirichlet polynomials with multiplicative coefficients, not just in the previously treated regime where N is a small power of T. For the class F(c), the paper establishes the lower bound with the precise shape √N exp((√2+o(1))√(log(T/N) log_3(T/N)/log_2(T/N))), uniformly in N between exp((logT)^(1/2+δ)) and √T. In the transition regime where logN = √(logT log_2 T) τ, it shows the bound holds for all completely multiplicative f with |f(n)|=1, with the constant A(τ+τ') defined through the exponential-integral relations τ=∫_A^∞ e^{-u}/u du and τ'=∫_A^∞ e^{-u}/u² du. The constant √2 appears beca","pith_inferences":["The positive-correlation condition in F(c) is used only to keep off-diagonal terms nonnegative; a natural test is whether the same bound survives without it, by handling sign changes with a different weighting.","The proof of Theorem 1.1 depends on an unstated support property of the resonator r — all primes in its support exceed λ², so all supported integers are odd and their distance-one neighbours carry weight zero. Verifying this property explicitly is a concrete step that would close the gap left by the text's analogy to the I1 estimate.","The same resonance-plus-GCD-sum scheme may extend to other objects, such as character sums or short Dirichlet polynomials, wherever a diagonal term can be isolated by rapid decay of a weight."],"forward_implications":["If correct, these bounds show that the maximum of |∑_{n≤N} f(n)n^{it}| is at least √N times an exponential factor growing with log(T/N) throughout the near-complete range exp((logT)^(1/2+ε)) ≤ N ≤ √T.","The range of N covered improves on earlier resonance-method results, which required N ≍ T^{C(N)} with C(N) growing like a small power of logN.","For f≡1, Theorem 1.1 gives a lower bound for the classical zeta sum max_{t≤T}|∑_{n≤N}n^{it}| in the transition regime that is stronger than the previous best.","The appearance of the constant √2, the same constant as in maximal GCD sums, indicates that extreme values of these Dirichlet polynomials and extreme GCD sums are governed by the same arithmetic mechanism.","The bound in Theorem 1.2 is uniform over the class F(c), meaning a single resonator works for all such coefficients f."],"fun_headline_variants":["Resonance method now covers whole range N≤√T","Improved Omega results for multiplicative Dirichlet polynomials","Wider range for large values of multiplicative Dirichlet sums","Exponential improvement for Dirichlet polynomials up to √T","Multiplicative Dirichlet sums: new records across N≤√T"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"For Theorem 1.1, the load-bearing step is the claim that the off-diagonal contribution to the second moment I2(R,T) is negligible; the text justifies this as 'similar to' the I1 estimate, but the phase-gap argument there only yields a factor 1/logT, and the claim is actually saved by an unstated property: the resonator's support contains only odd integers, so their neighbours at distance one have r=0.","fun_headline_variants_meta":{"raw":{"variants":["Resonance method now covers whole range N≤√T","Improved Omega results for multiplicative Dirichlet polynomials","Wider range for large values of multiplicative Dirichlet sums","Exponential improvement for Dirichlet polynomials up to √T","Multiplicative Dirichlet sums: new records across N≤√T"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001381,"raw_usage":{"total_tokens":5391,"prompt_tokens":667,"completion_tokens":4724,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":411,"completion_tokens_details":{"reasoning_tokens":4642}},"tokens_in":411,"tokens_out":4724,"duration_ms":35723,"temperature":1.0,"reasoning_tokens":4642,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T18:46:28.831721+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for the r defined in Theorem 1.1 and logN = √(logT log_2 T), the off-diagonal sum ∑_{k≤N}∑_{m,n≤x, km≠n} f(km) overline{f(n)} r(m)r(n) Φ(T/logT log(km/n)) and compare it with ∑_{m≤x} r(m)^2. If this ratio does not tend to 0 as T→∞, the proof of Theorem 1.1 fails; a numerical check for moderate T would already indicate whether the claimed negligibility holds.","supporting_citations":[],"review_version":1}