Chapter 20 The Bombieri–Vinogradov theorem (proof)
–\(f\)\(D_f(x; N,m)\)\(f\)\(f\)\(f\)\(\{1, \dots, x\}\text{,}\)\(|f|_2 = (\sum_n |f(n)|^2 )^{1/2}\text{.}\)\(\Delta \in (0,1]\text{,}\)\(m \in (\ZZ/N\ZZ)^*\text{.}\)\(\chi\)\(r\text{,}\)\(s\text{,}\)\(k\)\(K = \Delta^{-6}\text{.}\)\(k > K\)\(k \leq K\text{,}\)\(\ell r\text{;}\)\(\chi\)\(K = \Delta^{-6}\text{,}\)\(c>0\)\(f\)\(\{1, \dots, x\}\)\(g\)\(\{1, \dots, y\}\text{,}\)\(h = f \star g\)\(\chi\)\(N\text{.}\)\(N = rs\text{,}\)\(r\)\(\phi(rs) \geq \phi(r) \phi(s)\)\(r,s\text{,}\)\(\chi\)\(r\text{.}\)\(r\)\(R = \Delta^{-1}\text{.}\)\(r \leq R\text{,}\)\(g\)\(|g|\)\(f\)\(r > R\text{,}\)\(P \lt r \leq 2P\)\(f\)\(g\)\(P = R, 2R, \dots\)\(P > Q\text{,}\)\(r\)\(R^{-1}\)\(r\)\(s\)\(\log Q\text{,}\)\(y,z \geq 1\)\(n > z\text{,}\)\(x\text{,}\)\(y=z=x^{1/5}\)\(x^{1/5} \lt n \leq x\text{,}\)\(\Lambda_0(n)\)\(\Lambda_1(n)\)\(n \lt x^{1/5}\text{.}\)\(\Lambda_1\text{.}\)\(\Lambda_1(n)\)\(n \leq x\text{.}\)\(1 \leq n \leq x\)\(O(\delta^{-1})\)\(y \lt n \leq (1 + \delta)y\text{,}\)\(x^{1/5} \lt \delta \leq 1\)\(L,M\)\((1+\delta)^j\text{.}\)\(L,M\)\(x^{1/5} \lt L,M \lt x^{4/5}\)\(LM = x\text{;}\)\(n \lt x^{1/5}\)\((1+\delta)^{-1} x \lt n \lt (1+\delta)x\text{.}\)\(O(\delta N^{-1} x \log x)\text{.}\)\(L,M\)\(l,m\)\(L \lt \ell \leq (1+\delta) L, M \lt m \leq (1 + \delta) M\text{.}\)\(L,M\text{,}\)\(\Delta = (\log x)^{-A}\text{;}\)\(Q = \Delta x^{1/2}\text{,}\)\(L,M\text{,}\)\(\delta = \Delta^{1/2}\text{,}\)\(\Delta^{1/2} x (\log x)^3\text{.}\)\(\psi(x) = x + O(\delta x)\text{.}\)\(B(A) = 2A + 6\text{.}\)\(c>0\)\(f\)\(\{1, \dots, x\}\)\(ab \neq 0\text{,}\)Exercises Exercises
2.
Use \eqref{eq:vaughan} to deduce \eqref{eq:bv1}.3.
Prove \eqref{eq:bv2}.4.
Prove Theorem~\ref{T:key bdh} by imitating the proof of Theorem~\ref{T:deviation}.5.
Deduce Corollary~\ref{C:bdh} from Theorem~\ref{T:key bdh}.
Hint.
Rewrite the difference in terms of \(D_f\) and \(D_g\text{.}\)
