{"id":"fe041599-9484-4d42-9547-5ecf2e640991","arxiv_id":"1908.09503","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors extend the asymptotic counting of RSA integers to a wide range of the ratio r and derive a conditional interpolation formula for the bias in products of two primes.","lead":"This paper proves new asymptotic formulas for counting RSA integers, products of two primes p and q with q at most r times p, over a much wider range of the ratio r than previously known. It also gives a conditional formula for a known bias in the distribution of such products when r is close to x, interpolating earlier results by Dummit, Granville, and Kisilevsky and Moree and Saad Eddin.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4 is false and Theorem 1 fails at r=x/4, where the claimed o(1) error conflicts with the standard x/log x secondary term; Theorem 4 may survive but the written proof chain needs repair.","rationale":"The reader identified the same weak spot: Lemma 4 is false and the proof of Theorem 1 is invalid. I agree with that diagnosis, and I sharpen it: Theorem 1 itself is false at the endpoint r=x/4, because the claimed error is o(1) while the true discrepancy from the stated main term is of size x/log x. This is more than a missing proof step. However, the central theorem highlighted in the strongest claim is Theorem 4, and the available evidence suggests Theorem 4 may still be true. The unabsorbed O(r/log(4r)) remainder from Gr(4r) is dominated by the error term of Theorem 2 throughout the range where it matters, and near r=1 the remainder is O(1), which is even smaller. Thus a corrected proof that keeps this remainder should go through, and the main uniform formula does not appear to be contradicted. For this reason I do not move the verdict: a conditional acceptance with mandatory correction of Lemma 4 and Theorem 1 is the right outcome. The paper's bias discussion is honestly labeled conditional and is not the load-bearing part.","tokens_in":21764,"tokens_out":27720,"duration_ms":273997,"concrete_test":"Re-derive Theorem 2 starting from equation (17) without using Lemma 4: keep the Gr(4r) remainder (O(1) for r=O(1), O(r/log(4r)) for larger r) and check whether it is bounded by the stated error x log r/((log x)^2 log(x/r)) for all r in the range 1+exp(-c sqrt(log x)) <= r <= x/4. If the bound holds, Theorems 2 and 4 stand and only Theorem 1 and Lemma 4 need correction; if it fails at any r, Theorem 4 is unsupported. As a simple independent check, evaluate Theorem 1 at r=x/4 and compare with pi_2(x) = x(log log x - log log 2)/log x + o(x/log x), which already disproves Theorem 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the main interpolation result relies on Lemma 4, which is false, and on the resulting error estimate in Theorem 1. At r=x/4, Lemma 1 gives pi_2(x;x/4)=pi_2(x). Substituting r=x/4 into Theorem 1 makes the integral vanish and leaves pi_2(x) = x log log x / log x + O(x e^{-c sqrt(log x)}/(r log x)) = x log log x / log x + O(4 e^{-c sqrt(log x)}/log x), and the last term is o(1). But the standard Landau expansion, e.g. pi_2(x) = sum_{p <= sqrt x} (pi(x/p) - pi(p)) with pi(y) = y/log y + O(y/log^2 y), gives pi_2(x) = x(log log x - log log 2)/log x + o(x/log x), a nonzero secondary term of size x/log x. Thus Theorem 1 as stated is false. The origin is the expansion of Gr(4r) in the proof of Theorem 1: it leaves an unavoidable O(r/log(4r)) remainder, and Lemma 4 claims r/log 2r is bounded by x e^{-c sqrt(log x)}/(r log x), which already fails at r = sqrt x. Since Theorem 2 is derived from Theorem 1, the written proof chain for Theorem 4 is broken. However, the target error in Theorem 2 is O(x log r/((log x)^2 log(x/r))), which is large enough near r=x/4 (about x/log x) and much larger than the Gr(4r) remainder in the intermediate range, so Theorems 2 and 4 are likely salvageable by retaining the remainder instead of invoking Lemma 4. This is a serious but localized proof gap, not an evident falsehood of the central uniform formula.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the counting function \\(\\pi_2(x;r)\\) of RSA integers, that is, products \\(pq\\le x\\) of two primes with \\(p<q\\le rp\\). It claims refined asymptotic formulas for large \\(r\\) (Theorems 1 and 2), for \\(r\\) very close to 1 (Theorem 3), a uniform formula for \\(1+x^{-5/12}<r\\le x/4\\) (Theorem 4), and a conditional bias formula generalizing the Dummit--Granville--Kisilevsky result (Theorem 5). The methods are standard analytic number theory: the prime number theorem with exponential-error estimates, partial summation, and a short-interval average result of Zaccagnini. The central intended contribution is the interpolation between the Decker--Moree small-\\(r\\) formula and Landau's formula for \\(\\pi_2(x)\\).","tokens_in":22060,"tokens_out":29529,"duration_ms":286003,"significance":"If fully proven, Theorem 4 would be a clean uniform asymptotic over the whole non-trivial range of \\(r\\), and Theorem 5 would be a natural conditional generalization of the known bias for products of two primes. The paper is well organized, the conditional framework \\((P)\\) is clearly labeled, and the technical work on short intervals in Section 4 is careful. However, the current version contains a false supporting lemma (Lemma 4) that invalidates the proof of Theorem 1 and the written derivations of Theorems 2, 4, and 5. The failure appears localized: retaining the \\(r/\\log r\\) remainder that Lemma 4 was meant to remove still fits within the error term of Theorem 2 and within the relative error of Theorem 4, so the main uniform formula is likely correct. There is no circularity and no parameter fitting; the proofs are honest and standard.","major_comments":[{"comment":"Lemma 4 is false as stated. In the first case \\(1\\le r\\le\\sqrt{x}\\) the proof asserts \\(\\sqrt{x}\\ll x e^{-c\\sqrt{\\log x}}/(r\\log x)\\); at \\(r=\\sqrt{x}\\) this becomes \\(1\\ll e^{-c\\sqrt{\\log x}}/\\log x\\), which is false for all sufficiently large \\(x\\). The second case, \\(\\sqrt{x}<r\\le x/4\\), requires \\(r^2\\ll x e^{-c\\sqrt{\\log x}}\\), which fails at \\(r=x/4\\). Hence the lemma cannot be used to discard the remainder \\(O(r/\\log r)\\) appearing in the evaluation of \\(G_r(4r)\\).","section":"Section 2, Lemma 4"},{"comment":"Theorem 1 is false as stated. Substituting \\(r=x/4\\) and applying Lemma 1 gives \\(\\pi_2(x;x/4)=\\pi_2(x)\\). The integral from \\(4r\\) to \\(x\\) vanishes, while the error term is \\(O(x e^{-c\\sqrt{\\log x}}/(r\\log x))=O(e^{-c\\sqrt{\\log x}}/\\log x)=o(1)\\). Thus Theorem 1 would imply \\(\\pi_2(x)=x\\log\\log x/\\log x+o(1)\\). The standard Landau expansion contains a secondary term of size \\(x/\\log x\\) (for example, \\(\\pi_2(x)=\\frac{x}{\\log x}(\\log\\log x+M+o(1))\\) with the Meissel--Mertens constant \\(M\\)), so the claimed \\(o(1)\\) error is impossible. The source is the approximation \\(G_r(4r)=4r\\log\\log4r/\\log4r+O(r/\\log4r)\\) followed by the invalid use of Lemma 4.","section":"Section 1, Theorem 1"},{"comment":"Because Lemma 4 is false, the transition in the proof of Theorem 2 that absorbs \\(O(r/\\log4r)\\) into \\(O(x\\log r/((\\log x)^2\\log(x/r)))\\) is invalid. The proof of Theorem 4 then invokes Theorem 2 in the range \\(r_0<r\\le x/4\\), so the written proof of the main uniform result is incomplete. I note that the final statements of Theorems 2 and 4 may still be true: for \\(r=x/4\\), the uncancelled \\(O(r/\\log4r)\\) is \\(O(x/\\log x)\\), which is already within the stated error of Theorem 2 and within the relative \\(O(1/\\log\\log x)\\) error of Theorem 4. The authors should revise the proofs and statements accordingly, for example by keeping the \\(O(r/\\log r)\\) remainder in Theorem 1 and using a direct bound \\(r/\\log r\\ll x\\log r/((\\log x)^2\\log(x/r))\\) where needed.","section":"Section 3, proof of Theorem 2; Section 6, proof of Theorem 4"},{"comment":"The proof of Theorem 5 also invokes Lemma 4 in the estimate of \\(\\pi_{2,Q}(x;r)\\) near the end of the proof, when bounding \\(r/\\log2r\\) against the main term. Since Lemma 4 is false, this step needs an alternative argument; omitting it leaves a gap in the proof of Theorem 5 as well.","section":"Section 7, proof of Theorem 5"}],"minor_comments":[{"comment":"The integral in Theorem 1 is typeset ambiguously as \\(\\int_x^{4r}\\) in the plain text; the lower limit should be \\(4r\\) and the upper limit \\(x\\).","section":"Section 1, Theorem 1"},{"comment":"The notation \\(\\pi(x)\\) in assumption (P) conflicts with the standard prime-counting notation already used in the paper; consider writing \\(P_\\chi(x)\\) or \\(\\mathrm{Li}(x)+E_\\chi(x)\\).","section":"Section 1, assumption (P)"},{"comment":"Expressions like \\(\\log\\log xr - \\log\\log x/r\\) should be parenthesized as \\(\\log\\log(xr)-\\log\\log(x/r)\\) to avoid ambiguity.","section":"Throughout"},{"comment":"There are several typos: 'approixmate' in the proof of Theorem 3, 'provied' in the introduction, 'Vallée Pouusin' in Section 7, and the punctuation in 'Dummit, Granville and Kisilevsky'.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The false Lemma 4 is the central obstacle; the paper cannot be accepted in its current form. The evidence suggests the main uniform formula (Theorem 4) is salvageable, and I would encourage a revision that weakens Theorem 1 appropriately and replaces Lemma 4 with a direct estimate of the \\(O(r/\\log r)\\) remainder."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. The wide-range formula for RSA integers (Theorem 4) is a genuinely useful interpolation, and the conditional bias result (Theorem 5) is new. But the proof of these results as written relies on a false lemma, and Theorem 1 is actually false, not just unproven.\n\nWhat's new: Theorem 2 and 4 give an asymptotic formula for π2(x;r) over the whole range 1+x^{-5/12}<r≤x/4, interpolating Landau at r=x/4 and Decker-Moree at small r. That's a real contribution. The bias theorem with Lχ(s) is a natural generalization of Dummit-Granville-Kisilevsky and Moree-Saad Eddin, honestly labeled conditional on (P). The methods are standard PNT and short-interval estimates, used competently.\n\nWhere it goes wrong. Lemma 4 asserts r/log 2r ≪ x e^{-c√log x}/(r log x) for all r≤x/4. The proof's key step is √x ≪ x e^{-c√log x}/log x, which is false for large x (try r=√x). This is not a cosmetic gap. The proof of Theorem 1 uses Lemma 4 to absorb the Gr(4r) remainder O(r/log r) into the stated error term. At r=x/4, the claimed error term is o(1), while the remainder is O(x/log x). Indeed, Theorem 1 at r=x/4 says π2(x)=x log log x/log x + o(1), contradicting the standard secondary term of size x/log x. So Theorem 1 is false, not merely unproved. The reader's conditional verdict is too generous; the stress-test note is right to call it a load-bearing gap.\n\nWhat likely survives. Theorem 2's target error is O(x log r/((log x)^2 log(x/r))), which at r=x/4 is O(x/log x) and larger than the Gr(4r) remainder elsewhere. So Theorems 2 and 4 are probably salvageable by keeping the O(r/log r) term and changing the proof route. Theorem 5 depends on Theorem 4, so its proof needs the same repair. None of this looks like a false central idea; it looks like a local error in error-term bookkeeping.\n\nWho this is for: analytic number theorists interested in products of two primes or RSA-like counting functions. The paper deserves a serious referee, but the authors need to fix Lemma 4, restate or delete Theorem 1, and rework the proofs of Theorems 2, 4, 5. If they do that, the interpolation result would be a solid paper. I'd send it to peer review with a strong request for revision. I wouldn't cite it in current form.","headline":"Good ideas, broken endpoint: Theorem 1 is false at r=x/4, and the proof chain for the otherwise plausible Theorems 2 and 4 needs repair.","tokens_in":22701,"tokens_out":6990,"would_cite":false,"duration_ms":61575,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N25","11N69"],"pacs":[],"model":"deepseek-v4-flash","headline":"One formula now counts products of two primes with ratio bound across nearly all scales.","keywords":["RSA integers","products of two primes","asymptotic formula","distribution of prime numbers","Dirichlet characters","primes in short intervals","number theory"],"falsifier":"Check the proof's boundary estimate at $r=x/4$, where Lemma 4 asserts $r/\\log 2r \\ll x e^{-c\\sqrt{\\log x}}/(r\\log x)$; this reduces to $x/\\log x \\ll e^{-c\\sqrt{\\log x}}$, which is false for large $x$. Independent evidence could come from enumerating semiprimes up to a large fixed bound at $r=x/4$ and comparing with the claimed main term.","tokens_in":21460,"feed_emoji":"🔢","tokens_out":12092,"duration_ms":111003,"temperature":0.7,"pith_summary":"This paper tries to prove that the number of RSA integers—products $pq$ of two primes with $p<q\\le rp$—is governed by a single asymptotic formula throughout almost the whole range of the ratio $r$. The claimed formula reads\n$$\\pi_2(x;r)=\\frac{x}{\\log x}\\left(\\log\\log(xr)-\\log\\log\\frac{x}{r}\\right)\\left(1+O\\left(\\frac{1}{\\log\\log x}\\right)\\right)$$\nfor $1+x^{-5/12}<r\\le x/4$. A sympathetic reader would care because previous formulas worked only for small $r$ or only for large $r$, while this one bridges the regimes: at $r=x/4$ it reproduces the classical count of all products of two primes, and for $r$ close to $1$ it reduces to $2x\\log r/(\\log x)^2$. The paper also identifies where a known congruence bias appears for these products, namely only when $r$ is close to $x$. If the formula is right, it gives a uniform quantitative description of products of two primes under a ratio constraint.","feed_headline":"One formula counts RSA integers across nearly all ratios","feed_subtitle":"It bridges the small-ratio regime and the full product-of-two-primes regime in one theorem.","key_machinery":"The load-bearing identity is the dissection at $p=\\sqrt{x/r}$:\n$$\\pi_2(x;r)=\\sum_{p\\le\\sqrt{x/r}}\\pi(rp)+\\sum_{\\sqrt{x/r}<p\\le\\sqrt{x}}\\pi(x/p)-\\sum_{p\\le\\sqrt{x}}\\pi(p).$$\nIt splits the inner constraint into the region where $q\\le rp$ binds and the region where $pq\\le x$ binds. The main term is carried by a function $G_r(x)$ whose derivative with respect to $x$ is exactly $(\\log\\log(xr)-\\log\\log(x/r))/\\log x$, so integrating from $4r$ to $x$ produces the ratio-symmetric logarithm that appears in the theorem. For $r$ close to $1$, the proof uses a mean-square estimate for primes in short intervals, extended by taking a supremum over interval lengths, to control the error when $\\pi(rp)-\\pi(p)$ is replaced by its expected value $(r-1)p/\\log p$.","core_discovery":"The paper's central claim is that, for $1+x^{-5/12}<r\\le x/4$, the count $\\pi_2(x;r)$ of integers $pq\\le x$ with $p<q\\le rp$ equals\n$$\\frac{x}{\\log x}\\left(\\log\\log(xr)-\\log\\log\\frac{x}{r}\\right)\\left(1+O\\left(\\frac{1}{\\log\\log x}\\right)\\right),$$\nwith an absolute implicit constant. This uniform formula is assembled from two overlapping results: a large-$r$ formula valid down to $1+\\exp(-c\\sqrt{\\log x})$, and a small-$r$ formula valid up to $3/2$. The paper also proves a bias theorem under a uniform prime number theorem for Dirichlet characters: among RSA integers with $(pq,Q)=1$, the proportion with $\\chi(p)=\\chi(q)=\\eta$ is $\\tfrac14(1+H_{\\chi,\\eta}(x;r))$, where the bias term is controlled by a coefficient $L_\\chi(s)$ with $s=x/r$. This interpolates the previously observed bias for all products of two primes and the earlier no-bias conclusion for fixed $r$, and it indicates that the bias can appear only when $r$ is close to $x$.","pith_inferences":["Extension: the ratio-symmetric difference $\\log\\log(xr)-\\log\\log(x/r)$ is a natural measure of scale in the ratio variable, and the theorem could be inverted to estimate $r$ from observed counts of semiprimes, a direction the paper does not pursue.","Extension: the small-$r$ proof is limited by an exponent $5/12$ because of the available short-interval prime estimates, so improving those estimates should extend the uniform range even closer to $r=1$.","Extension: the paper's own discussion suggests a concrete numerical test of the bias threshold: compute $L_\\chi(s)$ for small $s$ and compare the predicted bias with exact counts at moderate $x$; this would substitute for the unproved lower-bound results on $L_\\chi(s)$.","Extension: because the proof absorbs a boundary term of order $r/\\log r$ when integrating $G_r(x)$ from $4r$, a fully repaired argument may need a slightly larger error term for $r$ near $x/4$; sampling the formula at $r=x/4$ would show whether the stated $O(1/\\log\\log x)$ relative error is plausible."],"forward_implications":["If the uniform formula is correct, the number of RSA integers is known up to a relative error $O(1/\\log\\log x)$ across the entire range $1+x^{-5/12}<r\\le x/4$.","At $r=x/4$, the formula reduces to the classical asymptotic for products of two distinct primes, so the two previously separate counting regimes are genuinely connected.","For $r$ close to $1$, the formula gives $\\pi_2(x;r)\\sim 2x\\log r/(\\log x)^2$, matching the earlier small-$r$ asymptotic while extending its range toward $r=1$.","Under the assumed prime number theorem for Dirichlet characters, the congruence bias has size tied to $L_\\chi(s)$; it is negligible for fixed $r$ and becomes potentially visible only when $r$ is within a small power of $x$."],"supporting_citations":[{"why":"introduces the RSA-integer counting function and supplies the small-r asymptotic baseline the new formula must match.","marker":"[1]"},{"why":"documents the congruence bias among products of two primes that the paper extends to the RSA-integer setting.","marker":"[2]"},{"why":"provides an earlier asymptotic formula valid for relatively large r, which the uniform formula refines.","marker":"[4]"},{"why":"are the classical asymptotic results for the total count of products of two primes, reproduced at r=x/4.","marker":"[5,6]"},{"why":"supplies the prime number theorem statements used to estimate the error terms throughout.","marker":"[7]"},{"why":"gives the integral representation of the count and the fixed-r no-bias conclusion the paper builds on.","marker":"[8]"},{"why":"provides the supremum version of the short-interval prime estimate needed for r close to 1.","marker":"[9]"},{"why":"gives the almost-all short-interval theorem that controls the error when r is very close to 1.","marker":"[10]"}],"fun_headline_variants":["RSA integer count formula now covers nearly all ratios","Bias in RSA integers emerges only when r approaches x","Wide-range RSA count formula bridges small and large r","One asymptotic formula unifies RSA integer counts across r"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof requires the boundary contribution at the lower cutoff $4r$—a term of size roughly $r/\\log r$—to be absorbed by the stated error term for every $r$ up to $x/4$, and the specific estimate used to do this fails in the upper part of that range.","fun_headline_variants_meta":{"raw":{"variants":["RSA integer count formula now covers nearly all ratios","Bias in RSA integers emerges only when r approaches x","Wide-range RSA count formula bridges small and large r","One asymptotic formula unifies RSA integer counts across r"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001068,"raw_usage":{"total_tokens":4496,"prompt_tokens":984,"completion_tokens":3512,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":3447}},"tokens_in":600,"tokens_out":3512,"duration_ms":24159,"temperature":1.0,"reasoning_tokens":3447,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:11:59.595751+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the proof's boundary estimate at $r=x/4$, where Lemma 4 asserts $r/\\log 2r \\ll x e^{-c\\sqrt{\\log x}}/(r\\log x)$; this reduces to $x/\\log x \\ll e^{-c\\sqrt{\\log x}}$, which is false for large $x$. Independent evidence could come from enumerating semiprimes up to a large fixed bound at $r=x/4$ and comparing with the claimed main term.","supporting_citations":[{"cited_title":"Decker and P","cited_arxiv_id":null,"evidence_quote":"introduces the RSA-integer counting function and supplies the small-r asymptotic baseline the new formula must match."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"documents the congruence bias among products of two primes that the paper extends to the RSA-integer setting."},{"cited_title":"Justus, On integers with two prime factors, Albanian J","cited_arxiv_id":null,"evidence_quote":"provides an earlier asymptotic formula valid for relatively large r, which the uniform formula refines."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the prime number theorem statements used to estimate the error terms throughout."},{"cited_title":"Moree and S","cited_arxiv_id":null,"evidence_quote":"gives the integral representation of the count and the fixed-r no-bias conclusion the paper builds on."},{"cited_title":"Saffari and R","cited_arxiv_id":null,"evidence_quote":"provides the supremum version of the short-interval prime estimate needed for r close to 1."},{"cited_title":"Zaccagnini, Primes in almost all short intervals, Acta Arith","cited_arxiv_id":null,"evidence_quote":"gives the almost-all short-interval theorem that controls the error when r is very close to 1."}],"review_version":1}