Sieving intervals and Siegel zeros

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corollary_1: Granville's corollary that, if there are infinitely many Siegel zeros, then for each fixed v > 1 some arbitrarily long intervals of length y = z^v have (F(v)+o(1))G(z)y integers free of primes up to z and others have (f(v)+o(1))G(z)y.

corollary_2: Granville's corollary that infinitely many Siegel zeros with 1 - beta < 1/(log q)^B, for some integer B >= 1, give infinitely many primes p_n with p_{n+1} - p_n >> log p_n (log log p_n)^{B-1}.

corollary_3: Granville's corollary that, if there are infinitely many Siegel zeros, then for arbitrarily large y there are admissible sets of length y with asymptotically 2y/log y elements, against the belief that y/log y is the largest possible size.

proposition_1: Granville's proposition that, along an infinite sequence of exceptional zeros, there are y and X for which the integers in (X, X+y] with no prime factor up to z, for y^{1-eps} > z > y^{1/2-o(1)}, number at most about (4y/(log y)^2) log^+(qy/z^2) + (1-beta_q)y.

proposition_2: Granville's proposition that an infinite sequence of exceptional zeros gives intervals of length y with at least 2y/log y minus an explicit loss of integers free of primes up to z, the loss depending on how close beta is to 1, in four regimes.

remark_p4: Granville's remark that the proof of his Corollary 2 shows that infinitely many Siegel zeros with 1 - beta < 1/(log q)^B, for some integer B >= 1, give integers m with J(m) >> omega(m)(log omega(m))^B, against the conjectured size omega(m)(log omega(m))^{3+o(1)} when B > 3.

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Andrew Granville, "Sieving intervals and Siegel zeros," Acta Arithmetica 205 (2022), 1--19, doi:10.4064/aa201002-25-6; first circulated as arXiv:2010.01211 (2020).

Cited edition: the arXiv v1 manuscript (stamp "arXiv:2010.01211v1 [math.NT] 2 Oct 2020" on p. 1), 15 pages, the only arXiv version (the arXiv record lists v1 alone); the locators below are its pages and labels. The published version was not read: IMPAN serves the journal by subscription, and the two scripted requests to impan.pl on 2026-09-22 (the volume 205 issue 1 listing and the DOI-shaped path) both answered HTTP 403 with an empty body. Its section numbering, page numbers and any revised statements or constants are therefore not recorded here, and nothing below is keyed to it. Provenance of the copy read: downloaded from https://arxiv.org/pdf/2010.01211v1 on 2026-09-22; 222,744 bytes. For the arXiv v1 manuscript, the arXiv record names arXiv's non-exclusive distribution license (arXiv:2010.01211), every other right reserved.

Read status. Claims checked against arXiv v1 for Corollary 1 (p. 3), Proposition 1 (pp. 3--4), Corollary 2 and the Jacobsthal remarks after it (p. 4), Corollary 3 and Proposition 2 (p. 5). Proof partially verified: the proof of Corollary 3 (p. 10) and the concluding calculation in the proof of Proposition 2 (pp. 12--13) were checked, but the earlier exceptional-zero prime-distribution estimates on which they depend (Corollaries 4 and 5) were not independently rederived; the proofs of Corollary 1 (pp. 9--10), Proposition 1 (pp. 11--12) and Corollary 2 (p. 13) were read for structure only. Result pages: corollary_1, proposition_1, corollary_2, remark_p4, corollary_3 and proposition_2.

Interval sifting and the linear-sieve barrier

Write

$$S(x,y,z)=\#\{n\in(x,x+y]:(n,P(z))=1\},\qquad P(z)=\prod_{p\le z}p.$$

The Jurkat--Richert linear sieve gives upper and lower functions $F(u)$ and $f(u)$ for $S$ when $y=z^u$. Granville proves conditionally that actual intervals can attain these abstract extremal bounds: if infinitely many Siegel zeros exist, Corollary 1 (p. 3) gives, for each fixed $v>1$, arbitrarily large $x,X,y,z$ with $y=z^v$ and

$$S(x,y,z)=(F(v)+o(1))G(z)y,\qquad S(X,y,z)=(f(v)+o(1))G(z)y.$$

For $1\le v\le3$, the upper extreme is $S(x,y,z)\sim2y/\log y$ (p. 5). Thus the factor $2$ in the linear-sieve upper bound is not merely an artifact of applying a general sieve to intervals: under the Siegel-zero hypothesis, genuine intervals attain it. This is the parity-barrier phenomenon relevant to admissible tuples.

Jacobsthal's function

The two paragraphs on Jacobsthal's function after Corollary 2 (p. 4) define $J(m)$ as the least $J$ such that every $J$ consecutive integers contain an integer coprime to $m$. For $m=P(z)$, $J(P(z))$ is therefore the least $y$ for which $S(x,y,z)\ge1$ for every $x$. Iwaniec's interval-sieve estimate gives $J(P(z))\ll z^2$, hence $J(m)\ll(\omega(m)\log\omega(m))^2$ for $m=P(z)$, and Iwaniec deduced that

$$J(m)\ll(\omega(m)\log\omega(m))^2$$

for every $m$. Granville states that his proof of Corollary 2 (which is itself a lower bound for prime gaps) shows that sufficiently close Siegel zeros would instead produce exceptionally large values of $J(m)$ (remark on p. 4): if there are infinitely many Siegel zeros with $1-\beta<(\log q)^{-B}$ for some integer $B\ge1$, then there are integers $m$ with

$$J(m)\gg\omega(m)(\log\omega(m))^B.$$

Jacobsthal's problem concerns the minimum number of survivors in an interval, whereas E1204 is reached from the maximum number of survivors. They are relevant to one another because both are extremal questions for the same quantity $S(x,y,z)$ and both expose the obstruction created by exceptional zeros; the Jacobsthal bounds do not themselves estimate $A(k)$.

Corollary 3: largest admissible sets

A set $H\subseteq[0,y]\cap\mathbb Z$ has length at most $y$, and is admissible when, for each prime $p$, some residue class modulo $p$ is absent from $H$ (p. 5). The paper notes that the largest admissible set of length $y$ is believed to have $\sim y/\log y$ elements. The standard sieve upper bound for an admissible set, which the manuscript does not state for admissible sets, is

$$\#H\le(2+o(1))\frac{y}{\log y}.$$

Corollary 3 (p. 5) states that, conditional on infinitely many Siegel zeros, there are arbitrarily large $y$ and admissible sets $H(y)$ of length $y$ such that

$$\#H(y)\sim\frac{2y}{\log y}.$$

The proof is on p. 10. Given $\epsilon>0$, Corollary 1 with $v=1/(1-\epsilon)$ supplies an $x$ and the set $B$ of $n\le y$ for which $x+n$ has no prime factor at most $y^{1-\epsilon}$, so that $B$ omits the class $-x$ modulo each such prime and $\#B\sim2y/\log y$. For each prime in $(y^{1-\epsilon},y]$, the proof deletes a least-populated residue class from the current set. The surviving proportion is at least

$$\prod_{y^{1-\epsilon}<p\le y}\left(1-\frac1p\right)\sim1-\epsilon.$$

The resulting set omits a residue class for every prime and has $(2+O(\epsilon))y/\log y$ elements. Letting $\epsilon\to0$ proves the corollary.

Proposition 2: quantitative approach to the barrier

Proposition 2 is on p. 5. It assumes an infinite sequence of exceptional zeros $\beta$ of real primitive characters of conductor $q$, and the print takes $z=y^u$ with $1\le u\le3$. The proof (p. 12) writes $x=z^u$, and the interval comes from $x=qy$ as in the proof of Proposition 1, so the card reads $u$ as the exponent with $z^u=qy$; the print does not reconcile the two. The proposition then gives values of $X$ with the following lower bounds: $S(X,y,z)\ge2y/\log y-(2\delta C(u)+o(1))y/\log y$, where $C(u)=\sqrt{2(1-\log^+(u-1))}$, when $1-\beta\le\delta^2/\log q$ for a fixed $\delta>0$; $S(X,y,z)\ge2y/\log y-C_\kappa(u)(\log y)^{2/(\kappa+1)}y/(\log y)^2$ for some constant $C_\kappa(u)>0$, when $1-\beta\le(\log q)^{-\kappa}$ for a fixed $\kappa>1$; $S(X,y,z)\ge2y/\log y-c_\tau(\log\log y)^\tau y/(\log y)^2$ for some constant $c_\tau>0$, when $1-\beta\le\exp(-(\log q)^{1/\tau})$ for a fixed $\tau\ge1$; and $S(X,y,z)\ge2y/\log y-(2/\epsilon+o(1))y\log\log y/(\log y)^2$, when $1-\beta\le q^{-\epsilon}$ with $\epsilon\to0$ slowly with $q$.

The proof is on pp. 12--13, headed "More than the proof of Proposition 2". It takes the $\chi(a)=-1$ case of the preceding almost-prime count, transfers it to an interval while removing least-populated residue classes, writes $y=q^A$ and $1-\beta=1/(B\log y)$, and balances the losses $1/A$ and $C(u)^2/(4B)$ by taking $A=(2/C(u))((1-\beta)\log q)^{-1/2}$ and $B=(C(u)/2)((1-\beta)\log q)^{-1/2}$. For this choice, with

$$\log y=\frac{2}{C(u)}\left(\frac{\log q}{1-\beta}\right)^{1/2},$$

it obtains some $X$ with

$$S(X,y,z)\ge \frac{2y}{\log y} -(1+o(1))\frac{4y\log q}{(\log y)^2}.$$

Substituting the four stated hypotheses on $1-\beta$ and expressing $\log q$ in terms of $y$ gives the four bounds.

Conditional consequence for E1204

Let $A(k)$ have the meaning in E1204, and let $y_j\to\infty$ be the sequence from Corollary 3. Set $k_j=\#H(y_j)$. Then

$$k_j=(2+o(1))\frac{y_j}{\log y_j},\qquad \frac{\log k_j}{\log y_j}\to1,$$

and the constructed set gives $A(k_j)\le y_j$. On the other hand, applying the sieve upper bound for admissible sets recorded above to an extremal $k$-element set of length $A(k)$ gives

$$k\le(2+o(1))\frac{A(k)}{\log A(k)}.$$

Since $A(k)\ge k-1$, this implies

$$A(k)\ge\left(\frac12-o(1)\right)k\log k.$$

Combining the two inequalities along $k=k_j$ yields

$$\boxed{\displaystyle \frac{A(k_j)}{k_j\log k_j}\longrightarrow\frac12.}$$

This conclusion is conditional on infinitely many Siegel zeros, whose existence is unknown. It therefore does not resolve E1204 unconditionally; it shows that the proposed asymptotic $A(k)\sim k\log k$ would fail under that hypothesis. The paper does not determine the mean-value quantity $B(k)$ asked for in E1204.

For Problem 855, the paper does not mention the inequality $\pi(x+y)\leq\pi(x)+\pi(y)$. Corollary 3's sets have about $2y/\log y$ elements in $[0,y]$, about twice $\pi(y)$; the route from dense admissible sets to a failure of the inequality, recorded on the problem page, also needs the prime $k$-tuples conjecture, and the paper proves nothing about the inequality.

Bears on. #1204: Corollary 3, conditional on infinitely many Siegel zeros, gives admissible sets of $\sim2y/\log y$ elements in $[0,y]$; the inversion above, made in the corpus and not in the paper, turns this into $A(k_j)/(k_j\log k_j)\to1/2$ along a sequence, so $A(k)\sim k\log k$ would fail under that hypothesis. Nothing is said about $B(k)$. #855: Corollary 3's sets have about twice $\pi(y)$ elements; the paper does not mention the inequality, and the route to a failure of it also needs the prime $k$-tuples conjecture. #4: under the hypothesis of Corollary 2 with $B\ge2$, its gaps $\gg\log p_n\log\log p_n$ exceed the problem's bound for every $C$ (an observation made here); the problem is already settled unconditionally, and this adds nothing to that standing. #970: the remark on p. 4 gives, under infinitely many Siegel zeros with $1-\beta<(\log q)^{-B}$, integers $m$ with $J(m)\gg\omega(m)(\log\omega(m))^B$, a lower bound for the problem's $h(k)$ at $k=\omega(m)$ far below $k^2$; it decides neither question.

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