Proof-route overview

If \(z{\lt}y\) and every corrected candidate complement is below \(y^3\), then \(|\mathcal C(N)|-(P(N)+T(N))/2\le |\mathcal C(N)\cap G(N)|\). The proof sums the pointwise weight bound.

For every sufficiently large even \(N\), \(0.67\, \mathfrak S_{\mathrm{Liu}}(N)N/(\log N)^2\le |G(N)|\), where \(G(N)\) is the original set of primes \(p{\lt}N\) with \(N-p\ge 2\) and at most two prime factors counted with multiplicity.

Under the canonical coprime theorem, for every \(0{\lt}\epsilon \le 10^{-10}\) and \(A{\gt}0\) there is \(C{\gt}0\) such that eventually for even \(N\), \(T(N)\le 3.94033\mathcal X_N+CN/(\log N)^A\). Both cutoff and paper-modulus residuals are included.

The distinct source weighted lower bound implies \(W(N)\ge (2.6408-\eta )\mathcal X_N\) eventually for even \(N\), for every \(\eta {\gt}0\). Source-boundary and valuation losses are absorbed using \(\mathfrak S_{\mathrm{Liu}}(N)\ge U{\gt}0\).

For every \(A{\gt}0\) there are \(B\ge 0\), \(C{\gt}0\) and \(N_0\) such that \(\sum _{1\le q\le Q_B(N)}E^*(N,q)\le CN/(\log N)^A\) for integers \(N\ge N_0\), where \(Q_B(N)=\lfloor N^{1/2}/(\log N)^B\rfloor \) and \(E^*(N,q)=\max _{y\in \{ 0,\ldots ,N\} }\max _{0\le a{\lt}q,\, (a,q)=1}|\pi (y;q,a)-L_*(y)/\varphi (q)|\). Here \(L_*(x)=2/\log 2+\int _2^xdt/\log t\) for \(x{\gt}1\), while \(L_*(0)=L_*(1)=2/\log 2\) by the totalized interval-integral convention, not a principal value; \(E^*(N,0)=0\).

\(\forall \epsilon ,A{\gt}0\ \exists C{\gt}0,B\ge 0,N_0:\) the canonical \(L_2\) coprime weighted remainder majorant is \(\le CN/(\log N)^A\) for \(N\ge N_0\).

\(\forall U{\gt}0\ \exists C{\gt}0,B\ge 0,N_0\ \forall N\ge N_0,I,a:\ \sum _{1\le q\le Q_B(N)}M_{a,I,*}(N;q)\le CN/(\log N)^U\), provided \(|a|\le 1\), \(A_2\le N^{2/3}\), \((\log N)^{2B}\le A_1\).

For every fixed \(\varepsilon {\gt}0\), there is \(N_0\in \mathbb N\) such that, for all \(N\ge N_0\) and every real \(a\), \(\sum _{n\in \mathcal A}w_a(n)\le D_{1,19/10}(N)\). The threshold precedes \(a\); no evenness hypothesis is needed.

For every \(\delta {\gt}0\) and every fixed \(0{\lt}\varepsilon \le 1\), there exists \(N_0(\delta ,\varepsilon )\ge 4\) such that all even \(N\ge N_0\) satisfy \(G_{11}\le (10191/100000+\delta )\mathcal X_N\) on the original domain \(z\le r\le q\le s\le t\le B\). No factor \(1-\varepsilon \) is asserted: this bound uses the full upper-endpoint rough mother.

For every \(\delta {\gt}0\), there is \(\varepsilon _0\in (0,2/15]\) such that, for every fixed \(0{\lt}\varepsilon {\lt}\varepsilon _0\), there exists \(N_0(\delta ,\varepsilon )\ge 4\) for which all even \(N\ge N_0\) satisfy \((515093/200000000-\delta )\mathcal X_N\le 4D_{1,19/10}(N)\). The threshold is not asserted uniformly over all small positive epsilon.

\(\exists K\in \mathbb N\), \(4\le K\), \(\forall N\ge K\), \(2\mid N\Rightarrow (1/2500)\mathcal X_N{\lt}D_{1,19/10}(N)\). Here \(D\) counts distinct primes \(p\) and \(\mathcal X_N=\mathfrak S_{\rm Liu}(N)N/\log ^2N\), with no extra factor two.

\(\exists K\in \mathbb N\), \(4\le K\), such that for all even \(N\ge K\) there exist \(p,r,q\in \mathbb N\) with \(p\le N\), \(p,q\) prime, \(r=1\) or \(r\) prime, \(N=p+rq\) and \(r^{10}\le q^9\).

For every \(\delta {\gt}0\), there exists \(\varepsilon _0\in (0,2/15]\) such that, for every fixed \(0{\lt}\varepsilon {\lt}\varepsilon _0\), there is \(N_0(\delta ,\varepsilon )\ge 4\) for which all even \(N\ge N_0\) satisfy \(G_6+G_7-G_9-G_{11}-G_{12}+(124341093/200000000-\delta )\mathcal X_N\le 4D_{1,19/10}(N)\), at the fixed exponents of this chapter.

Fix \(0{\lt}\varepsilon {\lt}2/15\). One threshold works for every \(1/18{\lt}a{\lt}b{\lt}(1-3b)/3{\lt}c{\lt}1/3\). With the corresponding cutoffs, every even \(N\) beyond it satisfies \(3G_1+G_2-4G_3-G_4-G_5+G_6+G_7-2G_8-G_9-G_{10}^{\mathrm{corr}}-G_{11}-G_{12}-1334N^{1-a}\le 4D_{1,19/10}(N)\). In the analytic application \((a,b,c)=(4/53,4/33,3/11)\).