{"id":"d14ce028-0775-4223-9f70-916f466712e3","arxiv_id":"2411.18903","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the first two Mertens error terms, the Riemann hypothesis is equivalent to the positivity of their integral from 2 to X for every X>2.","lead":"This paper shows that the Riemann hypothesis is equivalent to the condition that the average of the first two Mertens error terms, the difference between observed and predicted prime sums, is positive on every interval from 2 to X. It also extends this to primes weighted by Dirichlet characters and to arithmetic progressions, with explicit lists where the equivalence holds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main Theorem 1 is sound; the load-bearing gap is Theorem 2's Θ=1 case, which rests on the sketched Lemma 10 and Assumption 1. An independent re-derivation of Lemma 10's amplitude and exponent would settle it.","rationale":"The reader's weakest assumption identifies exactly the right load-bearing concern: Lemma 10 and Assumption 1 are the only parts of Theorem 2's Θ=1 proof that are not fully demonstrated. I reviewed Theorem 1 independently: the sufficiency proofs for E1 and E2 use the explicit formula and the bound ψ(x)−θ(x)>0.98√x, and the constants work for X≥10^8; the necessity direction via Landau's oscillation theorem is standard and appears sound. The extension to E3 for Θ<1 also checks out: Lemma 9's upper bound on the second moment makes the higher-order terms negligible compared to the first moment's Ω± oscillation. The remaining gap is specifically the Θ=1 regime, where the proof needs a lower bound on |Δ1| of size x^β/(γ^{2+ε} log x) and an upper bound on Δ2. The upper bound is supplied by Lemma 11 under Assumption 1, but the lower bound rests on the sketched Lemma 10. I also noticed a minor circular typo in the proof of Lemma 9: the text says 'apply Lemma 9' where the context clearly requires Lemma 8; this is a typo, not a substantive flaw. The computational sets D and Q are not central to Theorem 1 and appear reproducible from the supplied code. Overall, the main theorem is solid, and the conditional Theorem 2 for Θ=1 remains conditional on the verification of Lemma 10.","tokens_in":18613,"tokens_out":14771,"duration_ms":129657,"concrete_test":"Independently re-derive Lemma 10 by carrying out the full power-sum argument for g(x)=∫_x^∞(Π(t)−lg(t))/t² dt, starting from the Mellin identity (29). The check is to verify the exponent of γ0 in (28) and the existence of x1,x2 in every interval [X,X^{1+ε}] by reproducing Pintz's [17, Theorem 2] proof step-by-step with the kernel d/dx(s x^{-s+1} log x − x^{-s+1}). If the amplitude is x^{β0}/(γ0^{1+ε} log x0) rather than (28), or if the interval property fails, the Θ=1 argument collapses. For additional assurance, numerically evaluate Δ1 at large X using a known zero at height γ0 and compare with the claimed bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central equivalence for i∈{1,2} (Theorem 1) is carefully argued, and I do not find a fatal gap there. The load-bearing weakness is in Theorem 2 for Θ=1, where the sign-change conclusion for E3 depends entirely on Lemma 10 and Assumption 1. Lemma 10 is only sketched as an adaptation of Pintz [17], and the key amplitude in (28) is claimed to be x_j^{β0}/(γ0^{2+ε} log x_j). The extra factor γ^{-2}, which is what makes Δ1 dominate Δ2 near σ=1, comes from the Mellin kernel computation in (29) and from the heuristic that f(x)=∫_x^∞(Π−li)/t² dt gains an additional 1/γ relative to Pintz's π−li oscillation. If the correct exponent is 1+ε rather than 2+ε, or if the double-integral interchange sketched in Lemma 10 loses a factor, then the inequality −Δ1(x2)>Δ2(x2) in §5 fails by a large γ factor, and the Θ=1 case of Theorem 2 does not follow. Assumption 1 itself is not unreasonable; the genuine weak point is that Lemma 10's proof is not supplied at the same standard as the rest of the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the averaged error terms E_i(x) in the three Mertens theorems. Theorem 1 shows, for i=1,2, that RH is equivalent to the condition that ∫_2^X E_i(x) dx > 0 for every X>2: under RH this is proved from the explicit formula for ψ and partial summation identities, while the converse uses Landau's oscillation theorem and the known numerical positivity of E_i up to 10^8. Theorem 2 treats E_3: positivity on RH when the supremun Θ of the real parts of the zeros equals 1/2, infinitely many sign changes when 1/2<Θ<1, and a corresponding conditional statement when Θ=1 under Assumption 1 and a Pintz-type oscillation lemma. Theorems 3 and 4 give analogous equivalences for real primitive Dirichlet characters and for arithmetic progressions, with finite classifications of the exceptional conductors, obtained by explicit zero computations and a verification with LCALC/SageMath.","tokens_in":18903,"tokens_out":24490,"duration_ms":220693,"significance":"Conditional on the gaps noted below, the paper contains a striking result: a simple positivity condition on a single integral is equivalent to RH, with no fitted parameters; the constants B1, B_chi_d, and B_q are computed from the functional equation and from zero data. The proof of Theorem 1 is essentially self-contained and uses standard tools, and the paper provides reproducible code for the computational parts. The E_3 result is more delicate and is genuinely conditional: the Θ=1 case depends on Assumption 1 and on a lemma whose proof is only sketched. If the missing details can be supplied, the paper is a solid contribution; in its present form the Θ=1 part of Theorem 2 is not proved to the same standard as the rest of the paper.","major_comments":[{"comment":"The proof of Lemma 10 is only a sketch. The estimate (28) is load-bearing: the denominator γ0^{2+ε} is exactly what makes −Δ1(x2) dominate Δ2(x2) in the final comparison; if the correct denominator were γ0^{1+ε}, the displayed inequalities would not imply −Δ1>Δ2. The text states that the γ1^2 factor appears from differentiation inside I2, but the interchange in (29), the power-sum argument, and the effective constants C1(ε), C2(ε) are not supplied. Since the paper explicitly adapts Pintz [17] to a different function (a double integral rather than π−li), a citation to [17] is not enough. Please give a complete proof of Lemma 10, or state it as an additional hypothesis and mark the Θ=1 clause of Theorem 2 as conditional on Lemma 10 as well.","section":"§5, Lemma 10"},{"comment":"The proof of Lemma 11 is a one-sentence reduction to [18, Theorem 1]. The lemma supplies the upper bound for Δ2 that competes with Lemma 10, and the statement includes a 'finitely many zeros' caveat when Θ=1. The adaptation should be written out, at least to the extent of showing explicitly that the exceptional finite zeros do not alter the bound x/e^{2(1−ε)ω(x)}. As written, the Θ=1 case of Theorem 2 depends on two results whose proofs are not supplied at the same standard as the rest of the paper.","section":"§5, Lemma 11"}],"minor_comments":[{"comment":"The positive constant 2/log2 − li(2) is dropped in the subsequent bound for π(x)−li(x); for X≥10^8 the term is negligible, but the displayed inequality should either include it or bound it explicitly.","section":"§2, Eq. (19)"},{"comment":"The domination of the n≥3 terms in (27) relies on |E2(x)|<1 for all x>2, citing [22, Theorem 5]. Please restate the exact range and the numerical bound from that theorem so the reader can verify the claimed inequality.","section":"§4, inequality (27)"},{"comment":"The proof only handles E1; the E2 case is dismissed with 'again we only address E1 here.' Since Theorem 4 asserts the equivalence for both i=1 and i=2, the analogous identity for E2 and the analogue of (30) should be stated explicitly, together with the reason the constant B_q is the same.","section":"§7, Theorem 4"},{"comment":"The notation for the sums over zeros, for example 'βj ≥ 1/2 + δ', is cramped; please define the indexing set explicitly (zeros of ζ with real part β and imaginary part γ, with |γ| ≤ X^2).","section":"§4, proof of Lemma 9"},{"comment":"The tables list B_chi_d to three decimal places; the repository should include the full computed values and the exact version of LCALC used, so that the computational classification is fully reproducible.","section":"Tables 2 and 3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is well within the scope of an analytic number theory journal. I did not independently rerun the computations in §6–§7; the provided code is a plus. The decisive question for publication is whether Lemma 10 can be proved in full. If the author prefers, the Θ=1 case could be restated as a theorem conditional on Assumption 1 plus Lemma 10, but then the advertised Theorem 2 would be weaker than its current wording suggests. I would not object to publication once the missing details are supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main thing to know: Theorem 1 is a real result and the proof is in good shape. The equivalence of RH with positivity of the mean values of E1 and E2 is new and genuinely extends Johnston's pi-li theorem. I read through the sufficiency and necessity arguments and I don't see a gap. The paper deserves a serious referee. What is actually new: the E1/E2 equivalence, the conditional analysis for E3, and the character and arithmetic progression classifications. The computations of D and Q are backed by Sage code and use explicit zero-counting bounds; that part looks honest and reproducible. The asymptotic formula (2) with the constant B1 is a nice addition, and the localized positivity corollary follows cleanly. Soft spots: Theorem 2 for Theta=1 rests on Assumption 1 and on Lemma 10, which is only sketched as an adaptation of Pintz. The key amplitude in (28) has gamma^{-2-epsilon}; Pintz had gamma^{-1-epsilon} for pi-li. That extra factor is exactly what makes Delta1 dominate Delta2. Since the proof of Lemma 10 is a sketch and the double-integral interchange isn't fully written out, I would want an independent derivation of that exponent before relying on the Theta=1 case. The author is upfront about Assumption 1 being unsatisfactory, and the main theorem doesn't depend on it. The intermediate case 1/2 < Theta < 1 is fine because the second moment is lower order; the boundary Theta=1 is genuinely delicate. The computational classification also deserves a check, but that's a minor concern. The borderline B values in Tables 2 and 3 are close to 2 (around 1.98), so a small numerical error could flip membership in D. The method is sound, and the code availability helps, but a referee should verify a few of the borderline cases. Bottom line: Theorem 1 is solid and significant within the comparative prime number theory program. Theorem 2's Theta=1 case is a conditional side result that needs a fuller proof of Lemma 10. I'd send this to peer review and ask the referee to focus on Lemma 10 and the numerical verification.","headline":"Theorem 1 is a solid new equivalence between RH and positivity of mean Mertens error terms; the Theta=1 case for E3 rests on a sketched lemma that needs a full proof.","tokens_in":666,"tokens_out":994,"would_cite":true,"duration_ms":25354,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26","11N05","11Y35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Riemann hypothesis is equivalent to a positivity condition on Mertens error terms.","keywords":["Mertens' theorems","Riemann hypothesis","Generalized Riemann hypothesis","error terms","explicit formula","Landau's oscillation theorem","Dirichlet characters","arithmetic progressions"],"falsifier":"To test Theorem 1, compute the integral from 2 to X of E_1 or E_2 at very large X using the explicit formula under the assumption of RH; a single X with a non-positive value would refute the sufficiency direction. For the Θ=1 case of Theorem 2, one could construct or prove the existence of a zero-free region not representable by any function satisfying Assumption 1, which would block the required control of the second moment.","tokens_in":18410,"feed_emoji":"🔢","tokens_out":6340,"duration_ms":55023,"temperature":0.7,"pith_summary":"The paper claims that the Riemann hypothesis is equivalent, for the first two error terms in Mertens' theorems, to a clean positivity condition: the integral of the error term from 2 to any X must be positive. If true, the most famous conjecture in number theory becomes a statement about the average behavior of prime sums, and the same equivalence extends to the third Mertens error term under an additional zero-free-region regularity assumption. The paper also proves analogues for prime sums twisted by real Dirichlet characters and for primes in arithmetic progressions, and it identifies the complete finite sets of characters and moduli for which the equivalence survives. A sympathetic reader would care because it gives a new formulation of RH that makes no reference to zeros and a template for converting zero-location conjectures into sign conditions on averages.","feed_headline":"Riemann hypothesis equals positive Mertens error averages","feed_subtitle":"For the first two Mertens terms, a positive mean on every interval is exactly RH; the third term works under one condition.","key_machinery":"The argument is carried by the explicit formula for ψ(x) and the related formula for π(x)-li(x), which turn the integrated error terms into sums over nontrivial zeros of ζ(s). The necessity direction uses Landau's oscillation theorem: if a zero lies off the critical line, the zero sum forces the integral to take both signs. The sufficiency direction is quantitative: under RH, the total zero contribution is controlled by the constant B1=-2 ξ'/ξ(0)=0.0461..., and positivity follows because the dominant term exceeds all remainder terms. For the third Mertens error term, E3 is expanded as an exponential in E2, so the sign of its integral is decided by a race between the first moment of E2, positive by Theorem 1 under RH, and its second moment, which is bounded using a mean-square estimate of ψ(x)-x; in the Θ=1 case a lemma following [17] produces the required large oscillations.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1: for i=1,2, the Riemann hypothesis holds if and only if the integral from 2 to X of the ith Mertens error term is positive for every X>2. For the third Mertens error term, RH implies the same positivity, and if the supremum of the real parts of the nontrivial zeros satisfies 1/2<Θ<1, or Θ=1 under Assumption 1, then the integral changes sign infinitely often. The proof also yields precise asymptotics: with the normalization f_i(X) in (1), the limit inferior and limit superior of f_i(X) are 2-B1 and 2+B1, where B1=-2 ξ'/ξ(0)=0.0461..., so the positivity holds with a fixed numerical gap. For real primitive characters, GRH for L(s,χ_d) is equivalent to the positivity condition for exactly the 178 fundamental discriminants listed in Tables 2 and 3; for arithmetic progressions, GRH modulo q is equivalent to the positivity condition for a=1 exactly for the 24 moduli in Q.","pith_inferences":["This suggests a general heuristic: for many prime-counting error terms, a zero-location conjecture is equivalent to the positivity of the integrated error, and the dividing line is whether the zero-contribution constant stays below 2; testing this for other arithmetic functions, such as those from number fields, is a natural extension.","The fixed numerical gap 2-B1 = 1.9539... is wide enough that the positivity at moderate X might be verifiable computationally before reaching the asymptotic regime, even though the proof only guarantees it for all X under RH.","The Θ=1 case shows the difficulty is a moment problem: controlling the second moment of E2 via zero-free regions is as hard as controlling the first moment, so any future proof removing Assumption 1 would likely require a new second-moment estimate.","The finite sets D and Q can be viewed as an average-bias classification, and similar finite lists may exist for other L-functions whose analogue of Bχ can be computed explicitly."],"forward_implications":["If Theorem 1 is correct, RH is equivalent to a positivity statement that can be stated without mentioning zeros: every finite-interval mean of the first two Mertens error terms is positive.","Under RH, Corollary 1 gives a localized version: for any c < (2-B1)/(2+B1))^2 = 0.9548..., the integral from cX to X of each E_i is positive for all large X, so positivity persists on short intervals near X.","If RH fails with a zero of real part Θ>1/2, the integral of E3 oscillates in sign infinitely often, making the positivity phenomenon exactly a zero-location phenomenon.","For real primitive characters, the equivalence holds precisely for the 178 discriminants in D; outside D, assuming linear independence of zero ordinates, GRH implies that positivity eventually fails, so the list is a sharp arithmetic classification.","For arithmetic progressions modulo q with residue 1, the same sharp classification holds for the 24 moduli in Q; for other q, GRH plus linear independence forces the positivity to fail."],"supporting_citations":[{"why":"Supplies the explicit formula for ψ(x), the related formula for π(x)-li(x), and Landau's oscillation theorem, which are the backbone of both directions of Theorem 1.","marker":"[16]"},{"why":"Provides the verified positivity of E_i(x) on [2,10^8] and the θ-ψ comparison bounds needed to start the sufficiency argument.","marker":"[22]"},{"why":"Gives the oscillation lemma adapted in Lemma 10 that produces large positive and negative spikes of the first moment of E2, used in the Θ=1 case.","marker":"[17]"},{"why":"Provides the zero-free-region bound that controls the second moment of E2 in the Θ=1 case.","marker":"[18]"},{"why":"Supplies the mean estimates for ψ(x)-x that yield the second-moment bounds on E2 in Lemma 8.","marker":"[25]"},{"why":"Provides the explicit zero-counting estimate used to compute Bχd and to bound tails in identifying the finite set D.","marker":"[1]"},{"why":"Gives the analogous equivalence for π(x)-li(x) whose method is adapted here to Mertens error terms.","marker":"[11]"}],"fun_headline_variants":["RH iff first two Mertens error integrals stay positive","Mertens error positivity: RH for two, conditional for third","Positive Mertens error means imply RH for first two theorems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 1, that the sharp boundary of the zero-free region is a differentiable decreasing function whose derivative grows slowly, which is needed only for the Θ=1 case of Theorem 2; if no such function exists, that particular conclusion is unproved, though Theorem 1 is independent of it.","fun_headline_variants_meta":{"raw":{"variants":["RH iff first two Mertens error integrals stay positive","Mertens error positivity: RH for two, conditional for third","Positive Mertens error means imply RH for first two theorems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000892,"raw_usage":{"total_tokens":3819,"prompt_tokens":892,"completion_tokens":2927,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":508,"completion_tokens_details":{"reasoning_tokens":2871}},"tokens_in":508,"tokens_out":2927,"duration_ms":24880,"temperature":1.0,"reasoning_tokens":2871,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T10:46:18.501969+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test Theorem 1, compute the integral from 2 to X of E_1 or E_2 at very large X using the explicit formula under the assumption of RH; a single X with a non-positive value would refute the sufficiency direction. For the Θ=1 case of Theorem 2, one could construct or prove the existence of a zero-free region not representable by any function satisfying Assumption 1, which would block the required control of the second moment.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the explicit formula for ψ(x), the related formula for π(x)-li(x), and Landau's oscillation theorem, which are the backbone of both directions of Theorem 1."},{"cited_title":"Approximate formulas for some functions of prime numbers","cited_arxiv_id":null,"evidence_quote":"Provides the verified positivity of E_i(x) on [2,10^8] and the θ-ψ comparison bounds needed to start the sufficiency argument."},{"cited_title":"On the remainder term of the prime number formula I. On a prob- lem of Littlewood","cited_arxiv_id":null,"evidence_quote":"Gives the oscillation lemma adapted in Lemma 10 that produces large positive and negative spikes of the first moment of E2, used in the Θ=1 case."},{"cited_title":"On the remainder term of the prime number formula II. On a theorem of Ingham","cited_arxiv_id":null,"evidence_quote":"Provides the zero-free-region bound that controls the second moment of E2 in the Θ=1 case."},{"cited_title":"Asymptotic distribution of prime numbers in the mean","cited_arxiv_id":null,"evidence_quote":"Supplies the mean estimates for ψ(x)-x that yield the second-moment bounds on E2 in Lemma 8."},{"cited_title":"Counting zeros of Dirichlet L-functions","cited_arxiv_id":null,"evidence_quote":"Provides the explicit zero-counting estimate used to compute Bχd and to bound tails in identifying the finite set D."},{"cited_title":"On the average value of π(t) − li(t)","cited_arxiv_id":null,"evidence_quote":"Gives the analogous equivalence for π(x)-li(x) whose method is adapted here to Mertens error terms."}],"review_version":1}