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Statement

Setting. $S(x,y,z)$ counts the integers in $(x,x+y]$ with no prime factor up to $z$ (p. 1). The paper puts the result beside Iwaniec's lower bound, quoted on p. 3: if $y\gg z^2$ then $S(x,y,z)\ge\frac{4y}{(\log y)^2}(\log(y/z^2)-O(1))$.

Proposition 1 (pp. 3--4, quoted). "Suppose that there is an infinite sequence of primitive real characters $\chi$ mod $q$ such that there is an exceptional zero $\beta=\beta_q$ of each $L(s,\chi)$. For each, there exists a corresponding value of $y$ such that if $y^{1-\epsilon}>z>y^{1/2-o(1)}$ then there exists an integer $X$ for which

$$S(X,y,z)\lesssim\frac{4y}{(\log y)^2}\log^+(qy/z^2)+(1-\beta_q)y$$

where $\log^+t=\max\{0,\log t\}$. We can take $y=q^{A-1}$ with $A\to\infty$ as slowly as we like."

The statement does not define $\epsilon$; in the proof (p. 12) a given $\epsilon>0$ is met by taking $y=q^{1/\epsilon-1}$ and $\kappa=\epsilon^2$ in the Siegel-zero hypothesis, so that $\Delta=(1-\beta)\log qy\le\epsilon$. "Exceptional zero" is Landau's notion recalled on p. 6: a real zero $\beta\ge1-c/\log Q$ of $L(s,\chi)$ for a primitive real character modulo $q\le Q$.

Proof pointer

Pp. 11--12, under the heading "At the sifting limit, redux". From display (5) (p. 7), primes up to $x$ in a class $a$ modulo $q$ with $\chi(a)=1$ are scarce, about $(1-\beta)x/\phi(q)$ of them. Integers up to $x$ in such a class with no prime factor up to $z$ are primes when $z>x^{1/2}$, and primes or products of two primes when $x^{1/3}<z\le x^{1/2}$; the two-prime count is estimated by partial summation and the prime number theorem, giving the $\log^+(qy/z^2)$ term. With $x=qy$, the progression is turned into an interval as in the proof of Corollary 1.

Read depth

Claims checked: the statement was read clause by clause on the page images of arXiv v1 (pp. 3--4). The proof on pp. 11--12 was read for structure, not rederived. Nothing here is independently reviewed.

Dependencies

Lemma 1 and display (5) of the paper (p. 7), not given pages here; the change of variable of Corollary 1's proof. Used by Corollary 2.

Source. A. Granville, Sieving intervals and Siegel zeros, Acta Arith. 205 (2022), 1--19, doi:10.4064/aa201002-25-6; labels and pages are those of arXiv:2010.01211v1, the edition named on the source card.

Bears on

No problem page directly. It is the input to Corollary 2 and, through Corollary 2's proof, to the remark on Jacobsthal's function.