Chapter 13 Brun's combinatorial sieve
\(f: \NN \to \CC\)\(f\)\(P\)\(P\)\(f\)\(g\)\(d\)\(P\text{,}\)\(X = X(x)\)\(d\text{,}\)\(r_d(x)\)\(d\)\(x\)\(g(p) = 1\text{,}\)\(p\text{;}\)\(p\)\(P\text{.}\)\(z\)\(x\text{,}\)\(z \lt x^\alpha\)\(\alpha \in (0,1)\text{,}\)\(V(z)X\)\(R(x,z)\text{.}\)\(P\)\(z \geq x^{1/2}\text{,}\)\(S(x,z) = \sum_{p \leq x} f(p)\text{.}\)\(f\)\(n-2\)\(A>0\text{.}\)\(d\text{!}\)\(S(x,x^{1/2})\)\(x\text{,}\)\(S(x,x^{1/(N+1)})\)\(p\)\(p+2\)\(x^{1/(N+1)}\text{,}\)\(N\)\(R(x,z)\)\(D^+\)\(D^-\text{,}\)\(n\)\(P\text{,}\)\(D^+\)\(D^-\)\(\delta^-\)\(\delta^+\)\(D^+\)\(D^-\)\(\{1,\dots, y\}\)\(y\)\(x\text{.}\)\(\lambda^+(d)\)\(\lambda^-(d)\)\(\mu\)\(D^+\)\(D^-\text{,}\)\(D^+, D^-\)\(d\)\(d = p_1 \cdots p_r\)\(p_1 > \cdots > p_r\text{.}\)\(m\)\(m\)\(y_1, y_2, \dots\)\(d\text{.}\)\(1 \in D^{\pm}\text{.}\)\(V_n(z)\)\(g(p_1 \cdots p_n) V(p_n)\)\(p_1 > \cdots > p_n\)\(p_1 \lt z\text{;}\)\(p_n \geq y_n\text{;}\)\(p_m \lt y_m\)\(m \lt n\)\(m \equiv n \pmod{2}\text{.}\)\(n\text{,}\)\(P\)\(P(z) = n\)\(g(d) = 1\)\(d\text{.}\)\(\lambda^+\)\(\lambda^-\)\(y_1, y_2, \dots\text{.}\)\(y\)\(D^{\pm} \subset \{1, \dots, y\}\text{;}\)\(R(x,y) \geq |R^{\pm}(x,z)|\text{,}\)\(V^{\pm}(z)\text{.}\)\(y_i\)\(d\text{.}\)\(d = p_1 \cdots p_r\)\(p_1 > \cdots > p_r\text{;}\)\(\beta > 1\)\(D^+ \cup D^-\)\(\{1, \dots, y\}\)\(D^-\text{.}\)\(z \leq y\text{;}\)\(z = y^{1/s}\)\(s \geq \beta\text{.}\)\(g\text{.}\)\(K>1\)\(\kappa > 0\text{,}\)\(w,z\text{,}\)\(\kappa\)\(g\text{.}\)\(z\)\(y\text{,}\)\(V^+(z)\)\(V^-(z)\text{;}\)\(V_n(z)\text{.}\)\(p_1, \dots, p_n\)\(V_n(z)\text{,}\)\(m \lt n\text{,}\)\(m \equiv n \pmod{2}\text{,}\)\(m > 1\)\(m \not\equiv n \pmod{2}\text{,}\)\(m = 1\)\(m \not\equiv n \pmod{2}\text{,}\)\(m\)\(z > p_1 >\cdots > p_n \geq z_n\)\(\beta = \kappa b + 1\)\(1+x \leq e^x\)\(x = (\beta-1)^{-1} = 1/(\kappa b)\)\(1+x \leq e^x\)\(x = b (\log K)/n\)\(n! \geq e (n/e)^n\)\(a = b^{-1} e^{1+b^{-1}}\text{.}\)\(\beta > 1\text{,}\)\(b > 0\text{.}\)\(b\)\(a \lt 1\text{;}\)\(b = 9\)\(a \lt e^{-1}\text{.}\)\(p_1^{n+\beta} > y\text{.}\)\(p_1 \lt z = y^{1/s}\text{,}\)\(V_n(z) = 0\)\(n + \beta > s\text{.}\)\(y_1, y_2, \dots\)\(\beta = 9\kappa + 1\text{,}\)\(g(d)\)\(K\text{,}\)\(s \geq \beta\text{,}\)\(z = y^{1/s}\)\(n-2\)\(p\)\(p+2\)\(p\)\(p+2\)\(\leq x\)\(O(x /\log^2 x)\text{;}\)\(x\)\(x^{1/20}\)\(c x/\log^2 x\)\(c>0\text{.}\)\(R(x,z)\text{.}\)
