Chapter 22 Small gaps between primes (after Goldston-Pintz-Yıldırım)
\(p_n\)\(n\)\((p_{n+1}-p_n)/(\log p_n)\text{,}\)\(\pi(x+y) - \pi(x)\text{,}\)\(x\)\(y \sim \lambda \log x\text{,}\)\(\lambda\text{.}\)\(\epsilon \approx 0.24\text{.}\)\(\epsilon > 0\text{,}\)\(p_n\)\(p_{n+1} - p_n \lt \epsilon \log p_n\text{.}\)\(p_{n+1} - p_n \lt (\log p_n)^{1-\epsilon}\)\(\epsilon > 0\text{.}\)\(Q = x^\theta\)\(\theta > 1/2\text{.}\)\(c = c(\theta)\)\(p_n\)\(p_{n+1} - p_n \lt c\text{.}\)\(\theta > 20/21\text{,}\)\(c(\theta) = 20\text{.}\)\(k\text{;}\)\(k\)\(k\)\(\calH = (h_1, \dots, h_k)\)\(n\)\(n+h_1, \dots, n + h_k\)\(\max \calH - \min \calH\text{.}\)\(a(n)\)\(j=1,\dots,k\text{,}\)\(j\)\(x \lt n \leq 2x\text{,}\)\(n + h_1, \dots, n + h_k\)\(a(n)\)\(n\)\(n + h_1, \dots, n + h_k\)\(k\)\(R\)\(k\)\(x\)\(a(n)\)\(\rho\)\(\rho(1) = 1\)\(\{1, \dots, R\}\text{.}\)\(\rho\)\(\rho\)\(\ell\)\(k\text{,}\)\(C,c>0\)\(k, \ell\text{,}\)\(R \leq x^{1/2}/(\log x)^C\text{,}\)\(n+h_j\)\(n\)\([d_1, d_2]\text{,}\)\(g\)\(g(p) = v_{\calH}(p) - 1\text{.}\)\(C,c>0\)\(k, \ell\text{,}\)\(R \leq x^{1/2}/(\log x)^C\text{,}\)\(h \notin \calH\text{,}\)\(h \in \calH\text{,}\)\(k, \ell \to \infty\text{.}\)\(R \leq x^{1/2 - \epsilon}\text{,}\)\(R \leq x^{1/4+\epsilon}\text{,}\)\(R = x^{1/2 - \epsilon}\)\(R = x^{1/2 - \epsilon}\text{,}\)\(k = 7, \ell=1\text{.}\)\(\calH = \{11,13,17,19,23,29, 31\}\text{,}\)\(P(\log (R/d)/(\log R))\)\(P(1) = 1\)\(k\)\(0\text{.}\)\(R = x^{1/2 - \epsilon}\text{,}\)\(k = 6\text{,}\)\(\calH = \{7,11,13,17,19,23\}\)\(P\text{,}\)\((\log R)/(\log x)\)\([d_1, d_2]\)\(d_1, d_2 \leq R\text{,}\)\(R^2\text{.}\)\(\pi(x; N, m)\)\(p \leq x\)\(m\)\(N\text{.}\)\(A>0\)\(\epsilon>0\text{,}\)\(c>0\)\(Q = x^{1-\epsilon}\text{.}\)\(R = x^{1/2 - \epsilon}\text{;}\)\(Q = x^{1/2 - \epsilon}\text{.}\)\(R = x^{1/4 - \epsilon}\text{,}\)\(\calH\text{.}\)\(\calH\)\(a(n;\calH)\)\(a(n)\)\(\calH\text{.}\)\(\delta > 0\)\(n\)\(p_{n+1} - p_n \leq H = \delta \log x\text{.}\)\(n\)\(n+1, \dots, n+h\)\(\frakS(\calH)\)\(h \notin \calH\)\(h \in \calH\text{.}\)\(n+h\)\(d\)\((n+h_1)\cdots(n+h_k)(n+h)\)\((n+h_1)\cdots(n+h_k)\text{;}\)\(n+h > x > R\text{,}\)\(\rho(d) = 0\)\(d\text{.}\)\(k\)\(k+1\)\(\calH\)\(\calH,h\text{.}\)\(h \in \calH\)\(H/(\log x) = \delta\)\(h \notin \calH\text{.}\)\(R = x^{1/2 - \epsilon}\)\(k,\ell\)\((p_{n+1}-p_n)/(\log p_n)\text{.}\)\(N\)\(m\text{,}\)\(N+2, \dots, N+m\text{.}\)\(f\)\(p_{n+1} - p_n > f(p_n)\)\(n\text{?}\)\(P\)\(P(1) = 1\)\(k\)\((\log R)/(\log x)\)
