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Statement

Setting (pp. 1123--1126). $\mathcal H=\{h_1,\ldots,h_{k_0}\}$ is a fixed admissible set (see Theorem 1) with $k_0=3.5\times10^6$, a value the paper fixes from p. 1125 on. Further:

• $x$ is large, $\mathcal L=\log x$, and $n\sim x$ means $x\le n<2x$;
• $\theta(n)=\log n$ if $n$ is prime and $\theta(n)=0$ otherwise;
• $P(n)=\prod_{j=1}^{k_0}(n+h_j)$;
• $D=x^{1/4+\varpi}$ with $\varpi=1/1168$, so $D^2=x^{1/2+2\varpi}$;
• $D_1=x^{\varpi}$ and $\mathcal P=\prod_{p<D_1}p$, so $d\mid\mathcal P$ means $d$ is squarefree with every prime factor below $x^{\varpi}$;
• for $(d,c)=1$ and a sequence $\gamma$,

$$\Delta(\gamma;d,c)=\sum_{\substack{n\sim x\\ n\equiv c\ (\mathrm{mod}\ d)}}\gamma(n) -\frac{1}{\varphi(d)}\sum_{\substack{n\sim x\\ (n,d)=1}}\gamma(n);$$

• for $1\le i\le k_0$, $\mathcal C_i(d)=\{c:1\le c\le d,\ (c,d)=1,\ P(c-h_i)\equiv0\ (\mathrm{mod}\ d)\}$.

Theorem 2 (p. 1126). For $1\le i\le k_0$,

$$\sum_{\substack{d<D^2\\ d\mid\mathcal P}}\ \sum_{c\in\mathcal C_i(d)} \lvert\Delta(\theta;d,c)\rvert\ll x\mathcal L^{-A}. \tag{2.12}$$

Here, by the paper's conventions (pp. 1123--1124), $A$ is any sufficiently large positive constant and the implied constant depends at most on $\mathcal H$, $\varepsilon$ and $A$.

The level $D^2=x^{1/2+2\varpi}$ exceeds the level $x^{1/2-\varepsilon}$ that the Bombieri-Vinogradov theorem reaches, but the sum runs only over moduli free of prime factors $\ge x^{\varpi}$ and only over the residue classes $\mathcal C_i(d)$, not all reduced classes.

Source. Yitang Zhang, Bounded gaps between primes, Ann. of Math. (2) 179 (2014), no. 3, 1121--1174, DOI 10.4007/annals.2014.179.3.7, read in the journal's edition identified on the source card: the notation on pp. 1123--1126, Theorem 2 on p. 1126, its proof in Sections 6--14 (pp. 1143--1173).

Read depth. Claims checked: the statement and every symbol in it were read clause by clause against the notation of Section 2. The proof was not checked, and nothing here is independently reviewed.

Proof pointer

Sections 6--14, pp. 1143--1173, outlined by the paper on pp. 1126--1127. Moduli $d\le x^{1/2-\varepsilon}$ are handled by the Bombieri-Vinogradov theorem (p. 1143). For larger smooth moduli, a combinatorial identity for $\Lambda$ (Section 6, via Lemma 6) reduces the estimate to sums of $\lvert\Delta(\gamma;d,c)\rvert$ for Dirichlet convolutions $\gamma$ of three types. Because $d\mid\mathcal P$, such a $d$ factors as $d=rq$ with $r$ in a flexibly chosen range (Lemma 4), and this factorization drives every case. Types I and II (Sections 7--12) use the dispersion method of Fouvry-Iwaniec and Bombieri-Friedlander-Iwaniec, reducing to incomplete Kloosterman sums bounded by a variant of Weil's bound (Lemma 11). Type III (Sections 13--14) uses the Birch-Bombieri estimate from the appendix to Friedlander-Iwaniec (Lemma 12), which rests on Deligne's proof of the Weil conjectures, combined with the factorization to save a factor $r^{1/2}$.

Bears on

No problem page directly. The theorem is the input to Theorem 1, and through it to the companion series that Problem 15 records.