On the distribution of products of two primes

Sumaia Saad Eddin; Yuta Suzuki

REVIEW 4 major objections 4 minor 10 references

On the distribution of products of two primes

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read One formula now counts products of two primes with ratio bound across nearly all scales.

desk verdict 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. read the letter →

arxiv 1908.09503 v1 pith:7BYS6O4N submitted 2019-08-26 math.NT

Sumaia Saad Eddin , Yuta Suzuki This is my paper · ORCID

classification math.NT MSC 11N2511N69

keywords RSAintegersproductsoftwoprimesasymptoticformuladistributionprimenumbersDirichletcharactersinshortintervalsnumbertheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to prove that the number of RSA integers—products $pq$ of two primes with $p

What carries the argument

The load-bearing identity is the dissection at $p=\sqrt{x/r}$: $$\pi_2(x;r)=\sum_{p\le\sqrt{x/r}}\pi(rp)+\sum_{\sqrt{x/r}<p\le\sqrt{x}}\pi(x/p)-\sum_{p\le\sqrt{x}}\pi(p).$$ It 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 $(r-1)p/\log p$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $$\frac{x}{\log x}\left(\log\log(xr)-\log\log\frac{x}{r}\right)\left(1+O\left(\frac{1}{\log\log x}\right)\right),$$ with 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 $x$.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $x$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What's new: Theorem 2 and 4 give an asymptotic formula for π2(x;r) over the whole range 1+x^{-5/12}

Where 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.

What 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.

Who 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.

Referee Report

4 major / 4 minor

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 $\pi_2(x)$.

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 (4)

[Section 2, Lemma 4] 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 1, Theorem 1] 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 3, proof of Theorem 2; Section 6, proof of Theorem 4] 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 7, proof of Theorem 5] 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.

minor comments (4)

[Section 1, Theorem 1] 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 1, assumption (P)] 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 $\mathrm{Li}(x)+E_\chi(x)$.
[Throughout] Expressions like $\log\log xr - \log\log x/r$ should be parenthesized as $\log\log(xr)-\log\log(x/r)$ to avoid ambiguity.
[Throughout] 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'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's central asymptotic formulas are derived from the prime number theorem and standard analytic-number-theoretic lemmas, not from their own conclusions.

full rationale

This paper contains no circular derivation. The central results (Theorems 1–4) are obtained by decomposing the counting function π2(x; r), applying the prime number theorem with classical error terms, and evaluating the resulting integrals; there is no fitted parameter that is later relabeled as a prediction, and no target formula is used as an input to its own proof. The references to Moree and Saad Eddin [8] are self-citations in the sense that the first author is also an author of the present paper, but they are not load-bearing: formula (4) is quoted as motivational background, and the proof of Theorem 1 explicitly develops the argument afresh via Lemmas 11–13, while Theorem 5 is a conditional statement proved under the clearly stated hypothesis (P) and does not invoke the target result as an assumption. The reader's concern about Lemma 4 being false and potentially breaking the written proof chain is a mathematical correctness issue, not circularity: a false lemma can invalidate a proof without making it circular. For the circularity question, the derivation chain is self-contained and independently grounded in cited external results such as Montgomery–Vaughan and Zaccagnini. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters or invented entities. The paper relies on standard analytic number theory inputs: PNT, PNT for APs, Landau's semiprime count, and Zaccagnini's short-interval theorem. Theorem 5 is conditional on a generic PNT error hypothesis that is clearly stated.

assumptions (5)

standard math Prime number theorem with de la Vallee Poussin-type error term (Lemma 8)
Used throughout to estimate sums over primes; standard result from Montgomery-Vaughan.
standard math Prime number theorem for arithmetic progressions (Lemma 9)
Used in the proof of Theorem 5 to estimate sums of Dirichlet characters over primes.
standard math Zaccagnini's theorem on primes in almost all short intervals (Theorem 6)
Used in Lemma 14 and Lemma 15 to handle the case r very close to 1.
standard math Landau's asymptotic formula for products of two primes, π2(x) = x log log x / log x + O(x/log x) (equation (1))
Used as the baseline for the case r ≥ x/4 and to verify interpolation.
domain assumption Assumption (P) with conditions (Δ1), (Δ2), (Δ3) on the error term δ(x) for PNT in arithmetic progressions
Assumed for Theorem 5; includes GRH as a special case. The theorem is conditional on this generic error hypothesis.

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Cite this review

Pith. "Pith review of On the distribution of products of two primes." pith.science (2026). https://pith.science/paper/7BYS6O4N

@misc{pith2026190809503,
  author       = {Pith},
  title        = {Pith review of: On the distribution of products of two primes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7BYS6O4N}},
  note         = {Machine review of arXiv:1908.09503}
}

abstract

For a real parameter $r$, the RSA integers are integers which can be written as the product of two primes $pq$ with $p<q\leq rp$, which are named after the importance of products of two primes in the RSA-cryptography. Several authors obtained the asymptotic formulas of the number of the RSA integers. However, the previous results on the number of the RSA integers were valid only in a rather restricted range of the parameter $r$. Dummit, Granville, and Kisilevsky found some bias in the distribution of products of two primes with congruence conditions. Moree and the first author studied some similar bias in the RSA integers, but they proved that at least for fixed $r$, there is no such bias. In this paper, we provide an asymptotic formula for the number of the RSA integers available in wider ranges of $r$, and give some observations of the bias of the RSA integers, by interpolating the results of Dummit, Granville, and Kisilevsky and of Moree and the first author.

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Decker and P

A. Decker and P. Moree, Counting RSA integers, Results Math. 52 (2008), 35–39

work page 2008
[2]

D. Dummit. A. Granville and H. Kisilevsky, Big biases amo ngst products of two primes, Mathematika 62 (2016), 502–507

work page 2016
[3]

Ivi´ c, The Riemann Zeta-Function , John Wileys & Sons, 1985

A. Ivi´ c, The Riemann Zeta-Function , John Wileys & Sons, 1985

work page 1985
[4]

Justus, On integers with two prime factors, Albanian J

B. Justus, On integers with two prime factors, Albanian J. Math. 3 (2009), 189–197. 28 S. SAAD EDDIN AND Y. SUZUKI

work page 2009
[5]

Landau, Sur quelques probl` emes relatifs ` a la distri bution des numbres premiers, Bull

E. Landau, Sur quelques probl` emes relatifs ` a la distri bution des numbres premiers, Bull. Soc. Math. France 28 (1900), 25–38

work page 1900
[6]

Landau, ¨Uber die Verteilung der Zahlen, welche aus ν Primfaktoren zusammengesetzt sind, G¨ ott

E. Landau, ¨Uber die Verteilung der Zahlen, welche aus ν Primfaktoren zusammengesetzt sind, G¨ ott. Nachr. Math.-Phys. Kl. (1911), 361–381

work page 1911
[7]

H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I.Classical Theory , Cambridge University Press, 2007

work page 2007
[8]

Moree and S

P. Moree and S. Saad Eddin, Products of two proportional p rimes, Int. J. Number Theory 13 (2017), 2583-2596

work page 2017

Show all 10 references

[9]

Saﬀari and R

B. Saﬀari and R. C. Vaughan, On the fractional parts of x/n and related sequences. II, Ann. Inst. Fourier (Grenoble) 27 (2) (1977), 1–30

1977
[10]

Zaccagnini, Primes in almost all short intervals, Acta Arith

A. Zaccagnini, Primes in almost all short intervals, Acta Arith. 84 (3) (1998), 225–244. (S. Saad Eddin) Institute of Financial Mathematics and Applied Number Theory , Jo- hannes Kepler University, Altenbergerstrasse 69, 4040 Linz, Austria. E-mail address : sumaia.saad eddin@j...

1998

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Decker and P

A. Decker and P. Moree, Counting RSA integers, Results Math. 52 (2008), 35–39

work page 2008

[2] [2]

D. Dummit. A. Granville and H. Kisilevsky, Big biases amo ngst products of two primes, Mathematika 62 (2016), 502–507

work page 2016

[3] [3]

Ivi´ c, The Riemann Zeta-Function , John Wileys & Sons, 1985

A. Ivi´ c, The Riemann Zeta-Function , John Wileys & Sons, 1985

work page 1985

[4] [4]

Justus, On integers with two prime factors, Albanian J

B. Justus, On integers with two prime factors, Albanian J. Math. 3 (2009), 189–197. 28 S. SAAD EDDIN AND Y. SUZUKI

work page 2009

[5] [5]

Landau, Sur quelques probl` emes relatifs ` a la distri bution des numbres premiers, Bull

E. Landau, Sur quelques probl` emes relatifs ` a la distri bution des numbres premiers, Bull. Soc. Math. France 28 (1900), 25–38

work page 1900

[6] [6]

Landau, ¨Uber die Verteilung der Zahlen, welche aus ν Primfaktoren zusammengesetzt sind, G¨ ott

E. Landau, ¨Uber die Verteilung der Zahlen, welche aus ν Primfaktoren zusammengesetzt sind, G¨ ott. Nachr. Math.-Phys. Kl. (1911), 361–381

work page 1911

[7] [7]

H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I.Classical Theory , Cambridge University Press, 2007

work page 2007

[8] [8]

Moree and S

P. Moree and S. Saad Eddin, Products of two proportional p rimes, Int. J. Number Theory 13 (2017), 2583-2596

work page 2017