{"id":"b87070a4-9655-4a8c-9ae1-c238184db7b6","arxiv_id":"2412.17199","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under GRH, every sufficiently large even N has a+b=N with lambda(a)=lambda(b)=-1, and for prime N each of the four sign patterns occurs many times.","lead":"Assuming a standard unproven hypothesis about the distribution of primes (GRH), this paper proves that every large even integer can be written as the sum of two integers whose Liouville values are both -1. It conditionally answers a Goldbach-style question and introduces the Pierce expansion as a tool for binary sign-pattern problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.1's induction is arithmetically invalid: the recurrence g(p) <= 2p + sum_{r<p} 2^{r-1} g(r) with g(r) <= 2r^2 gives an exponential bound, so the key estimates in Proposition 2.1 are unsupported.","rationale":"The reader's weakest assumption (zero-free region for Dirichlet L-functions modulo N) is legitimate and is indeed needed in Lemma 3.3 and Proposition 2.2. However, the more immediate obstruction is internal: the proof of Proposition 4.1, which underpins the friable-number input to Proposition 2.1, contains an invalid induction. The recurrence derived from the Pierce-expansion counting has an exponential factor 2^{r-1} in front of g(r); with g(r) <= 2r^2 the sum is of size 2^p p^2, not < 2p^2. A check at p=5 already shows the claimed quadratic bound is not obtained from the printed argument, and the composite case (12) drops a factor (log R)^2. Therefore Lemma 5.2 and Propositions 5.1 and 2.1 are not presently supported. I do not claim the theorem is false; a weaker bound such as 2^q q^2 |E(N)| might still suffice in the final argument of Proposition 1.4, since q ~ (log log N)^6 and |E(N)| <= N exp(-C(log log N)^6). But the manuscript proves neither the stated Proposition 4.1 nor the weaker bound, and Remark 7 indicates the pointwise bound on nu_r(m) is a known bottleneck. A nontrivial repair is required before the central claim is established.","tokens_in":26191,"tokens_out":29049,"duration_ms":254236,"concrete_test":"Reproduce the induction for p=5 using the printed recurrence and Lemma 4.5: g(2)=2, g(3)<=6, g(4)<=8, so the bound for g(5) is <= 10 + 2*2 + 4*6 + 8*8 = 102, which does not imply the stated g(5) <= 50. If the recurrence or the values are not as printed, recompute the analogous bound for p=7; any correct evaluation of the printed induction will show the claimed quadratic bound fails. This settles whether Proposition 4.1 is currently proved.","verdict_should_be":"REJECT","load_bearing_attack":"The proof of Proposition 4.1 does not establish the stated bound. After (14) the author obtains the recurrence g(p) <= 2p + sum_{1<=r<p} 2^{r-1} g(r). Even granting the induction hypothesis g(r) <= 2r^2, the right-hand side is at least of order 2^p p^2, which cannot be <= 2p^2. Concretely, inserting g(2)=2, g(3)<=6, and g(4)<=8 from Lemma 4.5 gives g(5) <= 102, contradicting the claimed g(5) <= 50. The displayed chain '<= 2p + ... < 2p^2' is numerically false. The composite step in (12) likewise drops a factor of (log R)^2 when applying Lemma 4.5. Since Lemma 5.2 and Proposition 5.1 quantify |E_a(N)| using Proposition 4.1, the error term in Proposition 2.1 is not justified. The zero-free region is a real assumption, but the more immediate problem is internal: the quantitative engine of the paper is unproved as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript assumes GRH and claims that every sufficiently large even integer N has a representation N = a + b with λ(a) = λ(b) = -1, answering a problem of Shusterman. The main route is a quantitative sign-pattern count for prime N: Proposition 1.4 asserts that each of the four sign patterns occurs at least ≫ N exp(-C(log log N)^6) times. The proof proceeds by studying the minimal exceptional set E(N), proving an approximate dilation symmetry for Fourier coefficients of λ (Proposition 2.1), and contradicting it on average over primes via character sums under GRH (Proposition 2.2). The key intermediate is a recursive bound on the exceptional sets E_d(N) via Pierce expansions (Proposition 4.1), which is then combined with reciprocity (Lemma 4.3) and exponential sums over smooth numbers (Proposition 5.1).","tokens_in":26448,"tokens_out":15044,"duration_ms":134506,"significance":"If the technical content were correct, this would be a significant contribution: it would give a conditional answer to a natural analogue of the binary Goldbach problem under GRH, with an explicit quantitative lower bound for all sign patterns when N is prime. The proposed use of Pierce expansions and approximate dilation symmetry is novel and potentially useful for other binary problems. The paper is also careful about effectivity and about the possibility of weakening GRH to a zero-free region. However, the central recursive estimate, Proposition 4.1, is not established as written, and because the later Fourier-analytic argument depends on it, the main theorems are not proved by this manuscript.","major_comments":[{"comment":"The claimed induction is arithmetically false. The displayed recurrence g(p) ≤ 2p + Σ_{1≤r<p} 2^{r-1} g(r), combined with the induction hypothesis g(r) ≤ 2r^2, yields a right-hand side of size about 2^p p^2, not O(p^2). For example, with g(1)=0, g(2)≤2, g(3)≤6 and g(4)≤8 (the last from Lemma 4.5), the recurrence gives g(5)≤102, which contradicts the claimed g(5)≤50. Thus the universal bound g(d)≤2d^2 is unsupported.","section":"Section 4, proof of Proposition 4.1, Eq. (14)"},{"comment":"The composite step of the induction is also invalid as displayed. Substituting Ω(R)≤2 log R and P^+(R)≤R/2 gives Ω(R)^2 2P^+(R)^2 ≤ 2R^2(log R)^2, which is not ≤ 2R^2 for R≥4. Even if the prime case were repaired, the composite case would require a separate argument or a weaker allowed bound.","section":"Section 4, Eq. (12)"},{"comment":"Because Lemma 5.2 and Proposition 5.1 rely directly on Proposition 4.1 for the size of |E_a(N)| with q-friable a, the failure of Proposition 4.1 invalidates the exponential-sum estimate (15) and hence Proposition 2.1. The contradiction in the proof of Proposition 1.4 depends on Proposition 2.1, so the main theorems do not follow. The appendix's average bound on ν_r(m) cannot repair Eq. (14), since the induction needs pointwise control of ν_r(m) on the potentially sparse subset φ_p^{-1}(E_r(N))∩(N/(r+1),N/r), not merely on the full interval.","section":"Sections 5 and 6.1 (Lemma 5.2, Proposition 5.1, proof of Proposition 1.4)"}],"minor_comments":[{"comment":"The last line of the proposition reads 'there are 3 ≫ N e^{-C(log log N)^6} integers'; the symbol '3' appears to be a typo and should be '≥' or '≫'.","section":"Proposition 1.4"},{"comment":"The typeset text frequently collapses superscripts in the key estimates (for example '2(1+o(1))q2' and '22q2'), making it difficult to verify exact exponents; the manuscript should be typeset with unambiguous superscripts.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The gap in Section 4 is at the heart of the proof and is not cosmetic: the stated recursive bound for |E_d(N)| cannot be derived from Eq. (14) as written. Whether the paper can be repaired depends on a genuinely new amortized or pointwise estimate for the Pierce-expansion multiplicities ν_r(m). I would not accept the manuscript in its present form, but the overall strategy and the conditional result are potentially valuable if the missing estimate can be supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper addresses Shusterman's Goldbach-type problem for sign patterns of the Liouville function, conditionally on GRH. The main claims are genuinely new: prior work only forced mixed sign patterns, while here the author claims simultaneous (-,-) and (+,+) patterns for even N and quantitative lower bounds for all four patterns when N is prime. The Pierce-expansion mechanism is a fresh idea, and the overall architecture—approximate dilation symmetry, recursive control of exception sets, averaged character sums—is coherent and worth serious attention.\n\nThat said, the stress-test note is correct: the induction in Proposition 4.1 does not work. The recurrence (14) gives\n\ng(p) ≤ 2p + Σ_{1≤r<p} 2^{r-1} g(r).\n\nEven granting the induction hypothesis g(r) ≤ 2r², the sum is dominated by the r = p−1 term, which is of order 2^p p², not 2p². The displayed chain after (14) is numerically false; concretely, using g(2)=2, g(3)≤6, g(4)≤8 forces g(5) ≤ 102, contradicting the claimed g(5) ≤ 50. The problem is the pointwise bound ν_r(m) ≤ 2^{r−1}, which is far too crude. Since Lemma 5.2, Proposition 5.1, and ultimately Proposition 2.1 all rely on the friable bound g(r) ≪ (log r)² q², the quantitative engine is unsupported.\n\nTo be fair, the rest of the paper is thoughtful. The dependence on the author's own earlier paper [7] is legitimate—that is a published parameter-free proof. The zero-free-region remark is also credible: the theorems assume GRH anyway. And the appendix shows awareness that the 2^{r−1} bound is crude; a refined Pierce-expansion analysis might repair the proof. But as written, the claimed estimates do not follow, and the main theorems are not established.\n\nWho gets value from this? Specialists in analytic number theory who want to see a plausible route to Shusterman's problem and who might be able to fix the gap. It deserves a serious referee, not because the current proof is sound, but because the ideas are novel and the flaw looks potentially repairable. I would send it to review with a pointed request to verify Proposition 4.1, and I would not accept it in this form.","headline":"The paper targets a real open problem with a clever strategy, but the key recursive bound in Proposition 4.1 is arithmetically unsound as written, so the main theorems currently rest on an unjustified estimate.","tokens_in":26962,"tokens_out":4727,"would_cite":false,"duration_ms":40465,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N56","11N37","11M26","11P32"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves, conditional on the Generalised Riemann Hypothesis, that every sufficiently large even integer is a sum of two integers with Liouville value $-1$, via a new lower bound on the frequency of all four sign patterns of the…","keywords":["Liouville function","binary Goldbach problem","sign patterns","Generalised Riemann Hypothesis","Dirichlet L-functions","Pierce expansion","exponential sums over smooth numbers","zero-free region"],"falsifier":"Compute, for a large prime $N$, the exact counts of the four sign patterns of $(\\lambda(n),\\lambda(N-n))$ by factoring all $n<N$; if any pattern is absent, Theorem 1.3's second alternative is false at that $N$. For the proof's engine, locate a modulus $N$ and a non-principal character $\\chi$ with an $L$-function zero in the stated rectangle; then the estimate (8) used in Proposition 2.2 would be unavailable.","tokens_in":25994,"feed_emoji":"🔢","tokens_out":10741,"duration_ms":92688,"temperature":0.7,"pith_summary":"This paper attacks a binary Goldbach analogue: can every large even integer $N$ be written as $a+b=N$ with both $a,b$ having Liouville value $-1$? The author shows that, assuming the Generalised Riemann Hypothesis for Dirichlet $L$-functions, the answer is yes for every sufficiently large even $N$. The engine is a quantitative statement for primes: for each large prime $N$, all four sign patterns of $(\\lambda(n),\\lambda(N-n))$ occur at least $N e^{-C(\\log\\log N)^6}$ times. This is the first conditional resolution of the sign-pattern version of the problem, and the proof introduces a technique, built on the Pierce expansion of $n/N$, that is designed for pairs whose two entries are at very different scales.","feed_headline":"Every large even number is a sum of two Liouville-negative numbers","feed_subtitle":"Under the Riemann Hypothesis for Dirichlet L-functions, all four sign patterns appear many times.","key_machinery":"The argument is carried by the Pierce expansion of the rational number $n/N$, the alternating expansion $n/N=1/r_1-1/(r_1r_2)+1/(r_1r_2r_3)-\\cdots$ with strictly increasing integer denominators $r_j$. The paper encodes the failure of an exact dilation symmetry of the Fourier coefficients $S_\\lambda(a/N)=\\sum_{n<N}\\lambda(n)e(na/N)$ through the sets $E_d(N)=\\{n<N:\\lambda(dn)\\lambda(\\varphi_d(n))=-1\\}$, where $\\varphi_d(n)=N\\{dn/N\\}$. A recursion over Pierce signatures expresses membership in $E_p(N)$ for prime $p$ in terms of smaller $E_r(N)$, at a cost of $O(p|E(N)|)$ exceptions; combined with a reciprocity identity between $E_a(N)$ and $E_b(N)$ under $\\varphi$, this controls Fourier coefficients of $E_b(N)$. On the GRH side, averaged prime character sums force some $E_p(N)$ to be large for primes $p$ beyond $(\\log N)^{2+\\varepsilon}$, and the contradiction between the two estimates yields the sign-pattern lower bound.","core_discovery":"On its own terms, the paper's central claim is Theorem 1.2: under GRH there is an effectively computable $N_0$ such that every even $N\\ge N_0$ admits $1\\le a,b<N$ with $a+b=N$ and $\\lambda(a)=\\lambda(b)=-1$. The result is derived from the stronger Proposition 1.4, which asserts that for every sufficiently large prime $N$ and every pattern $(\\eta_1,\\eta_2)\\in\\{-1,+1\\}^2$ there are $\\gg N e^{-C(\\log\\log N)^6}$ integers $n<N$ with $\\lambda(n)=\\eta_1$ and $\\lambda(N-n)=\\eta_2$. The author is careful to record that full GRH is not needed: a zero-free rectangle of the form $\\operatorname{Re}(s)>1-(\\log N)^{-c}$ with $c<3/50$ and $|\\operatorname{Im}(s)|\\le(\\log N)^3$ would suffice, though with a weaker exponent.","pith_inferences":["If the appendix's average estimate for the preimage count $\\nu_r(m)$ could be upgraded to a typical or second-moment bound, the recursion would allow a much larger smoothness parameter $q$ and the contradiction would only need a Littlewood-type zero-free region; an unconditional theorem might then be within reach of zero-density methods.","The approximate dilation symmetry and Pierce-expansion encoding are not tied to $\\lambda$: the same architecture could plausibly establish sign-pattern frequencies for other real-valued multiplicative functions whose non-pretentiousness is available, especially in two-variable sums where the summands have very different sizes.","A direct computational check on primes $N$ up to any feasible bound, using exact values of $\\lambda$, would either confirm Proposition 1.4 in that range or, if a sign pattern is missing, give a concrete counterexample to the unconditional content of Theorem 1.3 for that $N$."],"forward_implications":["Under GRH, every sufficiently large even integer is a sum of two positive integers each having an odd number of prime factors counted with multiplicity.","For every sufficiently large prime $N$, all four sign patterns of the pair $(\\lambda(n),\\lambda(N-n))$ appear, and each appears many times: $\\gg N e^{-C(\\log\\log N)^6}$.","The evenness constraint in the binary-Goldbach analogue is only needed for the small friable residual cases; for any integer $N$ that has a prime factor larger than a computable $p_0$, all four sign patterns occur.","Because $N_0$ is effectively computable in terms of the parameter $p_0$, the remaining finitely many even $N$ below the threshold are in principle amenable to numerical verification.","The proof's zero-free-region requirement is much weaker than GRH, so the same theorem would hold under any future proof of the stated rectangle."],"supporting_citations":[{"why":"Supplies the initial bound $|L_\\lambda(N)|<N-1$, the dilation-symmetry rigidity argument that the recursion adapts, and the theorem about $\\lambda(M-d)=\\lambda(M+d)$ used in one small-friable case.","marker":"[7]"},{"why":"Provides the Pierce expansion framework, including the uniqueness convention and metric theory, on which the signatures $\\sigma_p(n)$ are based.","marker":"[15]"},{"why":"Gives the exponential-sum bound over smooth numbers that bounds Fourier coefficients of $E_b(N)$ in the minor-arc range.","marker":"[5]"},{"why":"Covers the complementary range of denominators where the other smooth-number exponential-sum theorem does not apply.","marker":"[3]"},{"why":"Supplies the smooth-number counting lower bound for $\\Psi(T,q)$ used when applying the two exponential-sum theorems.","marker":"[2]"},{"why":"Provides the discrepancy inequality used to convert Fourier-coefficient bounds into the uniform-distribution statement for $E_b(N)$.","marker":"[12]"},{"why":"Supplies the zero-counting and Perron-formula estimates for Dirichlet $L$-functions used in Lemma 3.2.","marker":"[13]"},{"why":"Provides the Hadamard-product and Perron-formula estimate for character sums over primes that Lemma 3.2 invokes.","marker":"[6]"},{"why":"Supplies the model averaged prime character sum estimate that Lemma 3.2 adapts.","marker":"[14]"}],"fun_headline_variants":["Goldbach for Liouville: Large evens are sums of two -1s","Under GRH every Liouville sign pattern appears often","Shusterman's problem: each sign pattern appears many times","Conditionally on GRH, binary Goldbach for Liouville negatives","Shusterman's Goldbach analogue: all sign patterns occur often"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that no non-principal Dirichlet character modulo $N$ has a zero in the rectangle $\\operatorname{Re}(s)>1-(\\log N)^{-c}$, $|\\operatorname{Im}(s)|\\le(\\log N)^3$ with $c<3/50$; if such a zero exists, the averaged prime character sum estimate (8) fails and the contradiction cannot be run.","fun_headline_variants_meta":{"raw":{"variants":["Goldbach for Liouville: Large evens are sums of two -1s","Under GRH every Liouville sign pattern appears often","Shusterman's problem: each sign pattern appears many times","Conditionally on GRH, binary Goldbach for Liouville negatives","Shusterman's Goldbach analogue: all sign patterns occur often"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001393,"raw_usage":{"total_tokens":5659,"prompt_tokens":990,"completion_tokens":4669,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":606,"completion_tokens_details":{"reasoning_tokens":4582}},"tokens_in":606,"tokens_out":4669,"duration_ms":30498,"temperature":1.0,"reasoning_tokens":4582,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:45:17.023477+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a large prime $N$, the exact counts of the four sign patterns of $(\\lambda(n),\\lambda(N-n))$ by factoring all $n<N$; if any pattern is absent, Theorem 1.3's second alternative is false at that $N$. For the proof's engine, locate a modulus $N$ and a non-principal character $\\chi$ with an $L$-function zero in the stated rectangle; then the estimate (8) used in Proposition 2.2 would be unavailable.","supporting_citations":[{"cited_title":"Mangerel","cited_arxiv_id":null,"evidence_quote":"Supplies the initial bound $|L_\\lambda(N)|<N-1$, the dilation-symmetry rigidity argument that the recursion adapts, and the theorem about $\\lambda(M-d)=\\lambda(M+d)$ used in one small-friable case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Pierce expansion framework, including the uniqueness convention and metric theory, on which the signatures $\\sigma_p(n)$ are based."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the exponential-sum bound over smooth numbers that bounds Fourier coefficients of $E_b(N)$ in the minor-arc range."},{"cited_title":"Fouvry and G","cited_arxiv_id":null,"evidence_quote":"Covers the complementary range of denominators where the other smooth-number exponential-sum theorem does not apply."},{"cited_title":"factorisatio numerorum","cited_arxiv_id":null,"evidence_quote":"Supplies the smooth-number counting lower bound for $\\Psi(T,q)$ used when applying the two exponential-sum theorems."},{"cited_title":"Montgomery","cited_arxiv_id":null,"evidence_quote":"Provides the discrepancy inequality used to convert Fourier-coefficient bounds into the uniform-distribution statement for $E_b(N)$."},{"cited_title":"Montgomery and R.C","cited_arxiv_id":null,"evidence_quote":"Supplies the zero-counting and Perron-formula estimates for Dirichlet $L$-functions used in Lemma 3.2."},{"cited_title":"Koukoulopoulos","cited_arxiv_id":null,"evidence_quote":"Provides the Hadamard-product and Perron-formula estimate for character sums over primes that Lemma 3.2 invokes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the model averaged prime character sum estimate that Lemma 3.2 adapts."}],"review_version":1}