{"id":"ea5d059c-59c1-4804-89c0-aa7c0901a545","arxiv_id":"2508.09750","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The maximum over characters of |Σ_{n≤N} f(n)χ(n)|, for any multiplicative f with |f(n)| = 1, is at least √N exp((1+o(1))√(log(q/N)/log₂(q/N))) in the range exp((log q)^{1/2+δ}) ≤ N ≤ √q.","lead":"This paper proves that weighted sums of Dirichlet characters can be very large: for any multiplicative f with |f(n)| = 1, some character χ gives |Σ_{n≤N} f(n)χ(n)| ≥ √N · exp((1+o(1))√(log(q/N)/log₂(q/N))) throughout exp((log q)^{1/2+δ}) ≤ N ≤ √q. It extends Hough's resonance-method lower bound for unweighted character sums to general multiplicative weights.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3's absorption of a merely multiplicative f into the resonator is invalid without complete multiplicativity; the theorem as stated is not established.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing gap: Section 3 uses |f(n)|=1 to absorb a multiplicative, not completely multiplicative, coefficient into the resonator, but the collapse along an=bm requires f(n)f(a)=f(na), which can fail at shared prime powers. The explicit tuple with f(p)=1, f(p^2)=-1 gives a phase -1 instead of 1, showing the asserted equality is not an identity. No defect-averaging argument appears in the visible text. Since this gap is central to Theorem 1.1 but may be repairable by adding a complete-multiplicativity hypothesis or a supplementary estimate, the Reader's CONDITIONAL verdict is appropriate; my stress-test does not change it.","tokens_in":3108,"tokens_out":13083,"duration_ms":147571,"concrete_test":"Verify the Section 3 replacement on the explicit tuple above: with p,ℓ in the support of r, define f(p)=1, f(p^2)=-1, f(pℓ)=f(ℓ)=1 (all other prime powers 1). Take N≥p^2 and X≥pℓ. Compute the contribution of (n,m,a,b)=(p^2,pℓ,ℓ,p) to the diagonal sum: the phase factor equals -1, whereas the asserted replaced summand is r(ℓ)r(p). If the equality is meant identically, this is a counterexample; if it is meant on average, the missing defect estimate must be exhibited.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central inequality rests on the diagonal replacement in Section 3: after setting r_f(n)=f(n)r(n), the text asserts 'Since |f(n)|=1' the diagonal becomes φ(q) Σ_{an=bm} r(a)r(b). The actual summand is f(n) overline{f(m)} f(a) overline{f(b)} r(a)r(b) (or f(n)f(m)f(a)f(b) in the OCR's dropped-bar notation). For this to collapse to r(a)r(b) along an=bm one needs f(n)f(a)=f(na) and f(m)f(b)=f(mb), i.e. complete multiplicativity. A merely multiplicative unit-valued f need not satisfy this when n and a share a prime. Explicitly, pick two resonator primes p,ℓ and define a multiplicative f by f(p)=1, f(p^2)=-1, f(ℓ)=f(pℓ)=1, all other prime-power values 1. For n=p^2, a=ℓ, m=pℓ, b=p we have an=bm=p^2ℓ but f(n) overline{f(m)} f(a) overline{f(b)} = f(p^2) overline{f(pℓ)} f(ℓ) overline{f(p)} = -1, not 1. Such tuples lie inside the required ranges for large q since p,ℓ are much smaller than N and X=q/N. Thus the claimed main term carries omitted phase defects, and no averaging estimate for these defects is supplied. The theorem as stated for all multiplicative f is not supported by the text; assuming complete multiplicativity would remove the gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims an Omega lower bound for character sums with multiplicative coefficients: for a prime q, an integer N with exp((log q)^{1/2+δ}) ≤ N ≤ √q, and any multiplicative f with |f(n)| = 1, there exists a nonprincipal character χ mod q such that |Σ_{n≤N} f(n)χ(n)| ≥ √N exp((1+o(1))√(log(q/N)/log₂(q/N))). The proof uses the resonance method: it defines a resonator r supported on squarefree integers with r_f(n)=f(n)r(n), expands the second moment over nonprincipal characters, reduces the congruence an≡bm mod q to the equality an=bm, and then invokes Hough's counting lemma to evaluate the diagonal sum.","tokens_in":3499,"tokens_out":12334,"duration_ms":146462,"significance":"If correct, the result is a natural and valuable generalization of Hough's large-character-sum lower bound to arbitrary unit-valued multiplicative weights, matching the unweighted extreme-value profile over the whole range N ≤ √q. The paper relies on external, independently established inputs (Hough's resonance counting lemma, the Granville–Soundararajan framework), and the proof strategy is coherent. However, the main theorem as stated is not established because a key diagonal simplification requires complete multiplicativity, which is not assumed.","major_comments":[{"comment":"The step 'Since |f(n)|=1, we can obtain M2 = φ(q)Σ_{an=bm} r(a)r(b) + O(qN)Σ|r_f|²' is invalid for a merely multiplicative f. The actual diagonal summand is f(n)overline{f(m)}f(a)overline{f(b)}r(a)r(b). Collapsing this to r(a)r(b) requires f(n)f(a)=f(na) and f(m)f(b)=f(mb), i.e. complete multiplicativity; the condition |f(n)|=1 alone is insufficient when n and a, or m and b, share a prime. For example, choose two primes p,ℓ in the resonator support and define a multiplicative f by f(p)=1, f(p²)=-1, f(ℓ)=1, and all other prime-power values 1. Take n=p², a=ℓ, m=pℓ, b=p; then an=bm=p²ℓ, all four variables are within range for large q, yet the diagonal summand equals -r(ℓ)r(p) instead of r(ℓ)r(p). Thus Hough's lemma is applied to an expression that has not been shown to equal the diagonal sum of r(a)r(b). No averaging estimate for the phase defects is supplied. The theorem as stated for all","section":"Section 3, diagonal step after defining r_f(n):=f(n)r(n)"}],"minor_comments":[{"comment":"The manuscript text contains severe encoding/rendering corruption, e.g. 'Theorem 1.1. ��� δ ∈ ...'. The formulas need to be carefully regenerated before the paper can be assessed or published.","section":"Theorem 1.1 and throughout"},{"comment":"The resonator weight is printed as r(p)=λ√q log p, and q is not introduced in Lemma 2.1. This is likely an OCR/typographical error (perhaps λ√p/log p); as printed, the dependence on q and the missing division are confusing and should be corrected.","section":"Lemma 2.1 and Section 3"},{"comment":"The assertion that the range exp((log q)^{1/2+δ}) ≤ N ≤ √q implies log N > 3λ log₂λ is made without derivation. A short justification would help the reader verify that Hough's lemma applies.","section":"Section 3, range condition"},{"comment":"After defining r_f(n):=f(n)r(n), the notation switches back to r for the counting lemma; the distinction between r_f and r should be maintained to avoid confusion in the diagonal simplification.","section":"Notation"},{"comment":"The abstract promises 'multiplicative coefficients', but the proof as written supports only completely multiplicative f. If the theorem is revised, the abstract and introduction should be aligned with the actual hypothesis.","section":"Abstract and introduction"}],"recommendation":"major_revision","confidential_remarks":"The central gap is real and load-bearing, but it is localized: replacing 'multiplicative' by 'completely multiplicative' would likely make the proof work, giving a weaker but still nontrivial extension of Hough's theorem. The manuscript is also unusually rough in presentation, with corrupt glyphs and ambiguous formulas, and needs a careful proofreading pass."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has a good target and the range arithmetic checks out, but the central diagonal step in Section 3 is wrong as written. The assertion 'Since |f(n)| = 1' does not allow replacing f(n)f(m)f(a)f(b) r(a)r(b) with r(a)r(b) on the diagonal an=bm. For merely multiplicative f, f(n)f(a) equals f(na) only when gcd(n,a)=1, and similarly for m,b. When gcd>1 there are phase defects, and those cases are not negligible in the stated range. The explicit counterexample in the stress-test note works: with n=p^2, a=l, m=pl, b=p, we get an=bm but the f-factor is -1, not 1. So the main term of M2 is not the clean expression claimed. The rest of the proof—Pólya–Vinogradov trace, Cauchy for the principal character, and the descent from Lemma 2.1—is standard and, conditional on that fixed diagonal, gives the stated lower bound. The result would be valid if f is assumed completely multiplicative, or if the authors can show the defect terms contribute only to the error term, which seems unlikely in this range because the resonator primes are small compared to N. The citation pattern is fine, and the use of Hough's lemma is appropriate. I'd send this to a referee, because the headline result is worth checking and the gap may be repairable by a small extra hypothesis, but as it stands Theorem 1.1 is not proven.","headline":"The claimed generalization to arbitrary multiplicative f fails at Section 3: |f|=1 does not make the diagonal phases disappear; the theorem needs complete multiplicativity or a nontrivial defect estimate.","tokens_in":4042,"tokens_out":5375,"would_cite":false,"duration_ms":57858,"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":"For every large prime modulus and any bounded multiplicative coefficient, some non-principal character makes the weighted sum $\\sum_{n\\le N} f(n)\\chi(n)$ at least $\\sqrt N\\exp((1+o(1))\\sqrt{\\log(q/N)/\\log_2(q/N)})$.","keywords":["character sums","multiplicative coefficients","large values","resonance method","Dirichlet characters","Omega results","second moment","prime modulus"],"falsifier":"Compute, for a moderately large prime $q$ with $N$ near $q^{1/3}$, the actual maximum over all non-principal $\\chi$ of $|\\sum_{n\\le N}\\lambda(n)\\chi(n)|$, where $\\lambda$ is the Liouville function; if for some $q$ this is consistently below $\\sqrt N\\exp((1+o(1))\\sqrt{\\log(q/N)/\\log_2(q/N)})$, the theorem's lower bound cannot hold as stated. A more direct check is to evaluate the diagonal sum in Section 3 for a non-completely multiplicative $f$ and see whether the missing phase factors alter the main term.","tokens_in":2963,"feed_emoji":"📐","tokens_out":11964,"duration_ms":116646,"temperature":0.7,"pith_summary":"This paper proves a lower bound for the largest possible size of a Dirichlet character sum whose terms carry an arbitrary multiplicative coefficient $f(n)$ with $|f(n)|=1$. The bound has the same shape as the unweighted case: for every sufficiently large prime $q$ and every $N$ with $\\exp((\\log q)^{1/2+\\delta})\\le N\\le \\sqrt q$, there is a non-principal character $\\chi$ such that $|\\sum_{n\\le N} f(n)\\chi(n)|\\ge \\sqrt N\\exp((1+o(1))\\sqrt{\\log(q/N)/\\log_2(q/N)})$. In other words, multiplying by a bounded multiplicative weight does not shrink the extreme values that character sums can attain. The proof uses the resonance method, choosing an auxiliary multiplicative weight so that a second-moment average over characters is dominated by the diagonal terms $an=bm$. The result extends the known $f(n)=1$ theorem to arbitrary bounded multiplicative coefficients and confirms a natural expectation in the range where character sums are hardest to pin down.","feed_headline":"Weighted character sums reach the unweighted record size","feed_subtitle":"Arbitrary bounded multiplicative coefficients cannot shrink the largest character sums, for every large prime modulus.","key_machinery":"The resonance method for character sums is the load-bearing device. An auxiliary multiplicative function $r(n)$ is chosen (squarefree-supported, with prime values set to a resonance profile on the interval $\\lambda\\le p\\le\\exp((\\log\\lambda)^2)$, $\\lambda=\\sqrt{\\log X\\log_2 X}$) so that the second moment of the product of the weighted character sum with its resonator concentrates on the diagonal. The key identity is the congruence reduction: because $NX\\le q$, the condition $an\\equiv bm\\pmod q$ collapses to the equality $an=bm$, and character orthogonality then produces the main term $\\phi(q)\\sum_{an=bm}r(a)r(b)$. Lemma 2.1 supplies the lower bound for this diagonal sum, which is what ultimat","core_discovery":"The paper's central claim is Theorem 1.1. It asserts that for fixed $\\delta\\in(0,1/100)$, for all sufficiently large primes $q$, all $N$ in the stated interval, and all multiplicative $f$ with $|f(n)|=1$, the maximum over non-principal characters of $|\\sum_{n\\le N}f(n)\\chi(n)|$ is at least $\\sqrt N\\exp((1+o(1))\\sqrt{\\log(q/N)/\\log_2(q/N)})$. The proof sets $X=q/N$ and studies the ratio $M_2/M_1$, where $M_2$ is the second moment over $\\chi\\ne\\chi_0$ of $|\\sum_{n\\le N}f(n)\\chi(n)\\sum_{a\\le X}f(a)r(a)\\chi(a)|^2$ and $r$ is a deliberately chosen multiplicative function. Character orthogonality and $NX\\le q$ turn the second moment into a diagonal sum over $an=bm$, which is then evaluated with a","pith_inferences":["Inference: the proof as written appears to use complete multiplicativity in the reduction to $\\sum_{an=bm}r(a)r(b)$; a reader extending this to all multiplicative $f$ should check whether the phase defects $f(na)/(f(n)f(a))$ average to $1$ on the diagonal.","Inference: the same diagonal orthogonality argument is naturally portable to other arithmetic weights, such as $n^{it}$ or short-interval twists, where the equality $an=bm$ is replaced by a different additive condition.","Inference: the resonance interval $\\lambda\\le p\\le\\exp((\\log\\lambda)^2)$ controls the achievable exponent; varying its length is a plausible route to sharpening constants or extending the range below $\\exp((\\log q)^{1/2})$.","Inference: a numerical experiment with the Liouville function at $N=q^{1/3}$ would give a concrete check of whether the diagonal main term survives without complete multiplicativity."],"forward_implications":["For any sufficiently large prime $q$ and any $N$ in the range, the lower bound holds for every multiplicative $f$ with $|f|=1$, not just for the constant function.","The exponent gained over $\\sqrt N$ is exactly $\\sqrt{\\log(q/N)/\\log_2(q/N)}$, matching the unweighted extreme-value profile throughout the interval.","The proof's diagonal reduction works whenever $NX\\le q$, so the whole range below $\\sqrt q$ is covered uniformly.","The theorem establishes the same lower-bound shape as the unweighted case, which is what the conjectural picture predicts for weighted character sums."],"supporting_citations":[{"why":"Provides the unweighted $f(n)=1$ lower bound and the resonance-weight construction that Theorem 1.1 generalizes.","marker":"[5]"},{"why":"Sets up the conjecture that maximal character sums increase up to $\\sqrt q$ and supplies earlier lower bounds in the ranges treated.","marker":"[2]"},{"why":"Gives the analogous extremal result for Dirichlet polynomials with $n^{it}$ coefficients, the closest recent comparison for the weighted problem.","marker":"[8]"},{"why":"Bounds low moments of weighted character sums and formulates the expectation that multiplicative coefficients do not reduce extreme values.","marker":"[3]"}],"fun_headline_variants":["Bounded coefficients can't shrink large character sums","Weighted character sums hit unweighted record size","Unit weights don't reduce character sum peaks","Multiplicative weights can't shrink character sum records"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The load-bearing premise is that, after character orthogonality, the diagonal contribution $\\sum_{an=bm} f(n)f(a)r(a)r(b)$ collapses to $\\sum_{an=bm} r(a)r(b)$, which requires $f$ to be completely multiplicative on those pairs while the theorem states only ordinary multiplicativity.","fun_headline_variants_meta":{"raw":{"variants":["Bounded coefficients can't shrink large character sums","Weighted character sums hit unweighted record size","Unit weights don't reduce character sum peaks","Multiplicative weights can't shrink character sum records"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001153,"raw_usage":{"total_tokens":4565,"prompt_tokens":642,"completion_tokens":3923,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":386,"completion_tokens_details":{"reasoning_tokens":3863}},"tokens_in":386,"tokens_out":3923,"duration_ms":28739,"temperature":1.0,"reasoning_tokens":3863,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T20:54:20.961977+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a moderately large prime $q$ with $N$ near $q^{1/3}$, the actual maximum over all non-principal $\\chi$ of $|\\sum_{n\\le N}\\lambda(n)\\chi(n)|$, where $\\lambda$ is the Liouville function; if for some $q$ this is consistently below $\\sqrt N\\exp((1+o(1))\\sqrt{\\log(q/N)/\\log_2(q/N)})$, the theorem's lower bound cannot hold as stated. A more direct check is to evaluate the diagonal sum in Section 3 for a non-completely multiplicative $f$ and see whether the missing phase factors alter the main term.","supporting_citations":[{"cited_title":"T he resonance method for large character sums, Mathematika, �� ,(2013), 87–118","cited_arxiv_id":null,"evidence_quote":"Provides the unweighted $f(n)=1$ lower bound and the resonance-weight construction that Theorem 1.1 generalizes."},{"cited_title":"Large character sums, J","cited_arxiv_id":null,"evidence_quote":"Sets up the conjecture that maximal character sums increase up to $\\sqrt q$ and supplies earlier lower bounds in the ranges treated."},{"cited_title":"E xtreme values of Dirichlet polynomials with multiplicative coefficients, J","cited_arxiv_id":null,"evidence_quote":"Gives the analogous extremal result for Dirichlet polynomials with $n^{it}$ coefficients, the closest recent comparison for the weighted problem."}],"review_version":1}