Chapter 14 The Selberg sieve
\(f: \NN \to \CC\)\(f\)\(P\)\(P\)\(f\)\(g(d)\)\(\lambda^+: \NN \to \RR\)\(V^+\)\(R^+\)\(\rho: \NN \to \RR\)\(\rho(1) = 1\)\(\rho\)\(\rho(1) = 1\text{,}\)\(\lambda^+(d) = 0\)\(d \geq y\text{,}\)\(y\text{;}\)\(\rho(n) = 0\)\(n \geq \sqrt{y}\text{.}\)\(\lambda^+\)\(L^2\)\(y\)\(y\)\(x\)\(f\)\(f\)\(x\text{.}\)\(R^+(z)\)\(V^+(z) x\)\(V^+(z)\)\(g(p) \in (0,1)\)\(p \in P\text{,}\)\(g(p) = 0\)\(p \notin P\text{.}\)\(g(p) \in [0,1)\)\(p \in P\text{,}\)\(P\)\(p\)\(g(p) = 0\)\(P\text{.}\)\(h\)\(c = \gcd(d_1, d_2)\text{,}\)\(a = d_1/c\text{,}\)\(b = d_2/c\)\(P(z)\)\(abc | P(z)\)\(\gcd(a,b) = 1\text{.}\)\(\gcd(a,b)\)\(e,f/e\)\(c,d\text{,}\)\(\xi\)\(\rho\text{.}\)\(\rho(1) = 1\)\(\rho(d) = 0\)\(d \geq \sqrt{y}\)\(\xi\)\(L^2\)\(f: \NN \to \RR_{\geq 0}\)\(P\)\(P(z) = \prod_{p \leq z, p \in P} p\text{.}\)\(d | P(z)\text{,}\)\(X>0\)\(g\)\(0 \lt g(p) \lt 1\)\(p \in P\text{.}\)\(h(d)\)\(h(p) = g(p) (1-g(p))^{-1}\)\(p \in P\text{,}\)\(y > 1\text{.}\)\(H\)\(d\)\(e\text{.}\)\(|\lambda^+(d)| \leq 3^{\nu(d)}\text{,}\)\(\nu(d)\)\(d\text{.}\)\(g\)\(c>0\text{,}\)\(\zeta^2(s)/\zeta(2s) =
\sum_{n=1}^\infty 2^{\nu(n)}n^{-s}\text{.}\)\(d(n)\)\(n\text{.}\)\(p \leq x\)\(O(x / \log^2 x)\text{.}\)\(f(n) = 1\)\(n = m(m+2)\)\(m\)\(f(n) = 0\)\(z = x^{1/4}\text{.}\)\(\epsilon > 0\text{,}\)\(x_0 = x_0(\epsilon)\)\(m,N\)\(\gcd(m,N) = 1\text{,}\)\(x \geq \max\{N, x_0(\epsilon)\}\text{,}\)\(p \leq x\)\(p \equiv m \pmod{N}\)\(\sum_n 1/(n \gamma(n))\)\(\gamma(n) = \prod_{p|n} p\text{.}\)\(d(n)\)\(n\text{.}\)
