3 Chen: a prime plus at most two primes

3.1 Chen’s theorem: from a signed sieve weight to representations

A prime \(p{\lt}N\) determines its partner \(N-p\) uniquely. We count representations by the prime \(p\), rather than by factorizations of its partner. The public finite set is

Thus \(|G(N)|=R_2(N)\) in the overview notation; \(\Omega (n)\) counts prime factors with multiplicity. Thus a prime partner and a square of a prime are both permitted, while \(N-p=1\) is excluded. This is exactly chenGoodRepresentations. The quantitative target is \(|G(N)|\ge 0.67 \mathcal X_N\) for all sufficiently large even \(N\), with the normalization

The second product is the positive constant \(U=C_2\) from the foundations chapter, and \(\mathfrak S_{\mathrm{Liu}}(N)\ge U{\gt}0\). This uniform lower bound will let us pay errors without choosing a different normalization for each even integer.

3.1.1 A finite weight that recognizes the desired partners

Put \(z=\max (2,\lfloor N^{1/10}\rfloor )\) and \(y=\lceil N^{1/3}\rceil \). The corrected candidate set is

Sifting has removed small factors, but a surviving partner can still have three or more prime factors. Let \(\mathcal Q_{z,y}\) be the primes in \([z,y)\). For a prime \(r\), let \(\nu _r(n)\) be its exponent in the factorization of \(n\). To charge these survivors, define

The second count records first factors \(r\), with the larger factors existentially quantified. In particular, its definition includes \(s=u\). For a \(z\)-rough integer \(1\le n{\lt}y^3\), meaning that no prime below \(z\) divides \(n\), positivity of \(w(n)\) forces \(n=1\), a prime, or a product of two primes. Indeed, the nonnegative integer \(v(n)+t(n)\) must be at most one. If \(v(n)=0\), all factors are at least \(y\), so three factors contradict \(n{\lt}y^3\). If \(v(n)=1\), there is exactly one medium factor, with valuation one; two remaining large factors would contribute to \(t(n)\) and use up the second unit.

For \(N\ge 9\) we have \(z{\lt}y\) and \(N\le y^3\); a candidate’s positive prime component gives the strict inequality \(N-p{\lt}y^3\). Consequently a bad candidate has penalty \(v(N-p)+t(N-p)\ge 2\). A good candidate has \(w(N-p)\le 1\), so summing the pointwise comparison \(w(N-p)\le \mathbf1_{G(N)}(p)\) gives

Here \(W\) is jurkatRichertWeightedCount; \(P,T\) are the corrected prime-power and triple penalties. The factor \(1/2\) comes from the bad-candidate inequality, not from a symmetry of a switched sum.

3.1.2 The source weight separates a lower sieve from upper sieves

The analytic lower bound first treats a slightly different finite carrier,

Let \(\mathcal Q(N)\) be the primes \(N^{1/10}{\lt}q\le N^{1/3}\), and set

Finite interchange of sums identifies this with the source weight \(\sum _{p\in \mathcal S(N)}(1-\frac12\# \{ q\in \mathcal Q(N):q\mid N-p\} )\). Each medium prime is charged once, irrespective of valuation. A lower bound for \(|\mathcal S(N)|\) and an upper bound for \(H(N)\) therefore advance the same signed inequality in the required directions.

For the base sieve, remove the exceptional unsifted primes whose complement is divisible by an odd cutoff prime dividing \(N\). Such an exceptional prime equals that divisor. The remaining sifting product and main mass are

Here \(\operatorname {li}_{\kappa }=L_{\kappa }\) in the foundations notation:

The latter is the implementation’s totalized-integral convention, not a Cauchy principal value. In particular \(X=L_*(N)\), where \(L_*=L_{2/\log 2}\) and \(L_*(0)=L_*(1)=2/\log 2\). See genuine logarithmic integral and finite-endpoint convention. The density is \(1/(r-1)\); \(R_N(d)=|\mathcal A_d|-X/\varphi (d)\) is the discrepancy of the unsifted complement carrier: \(\mathcal A_d\) consists of the remaining complements divisible by \(d\mid \mathcal P_N\). With level \(D=N^{1/2-\epsilon }\), lower Rosser coefficients give a finite lower bound by their density sum minus \(\mathcal R_0=\sum _{d\mid \mathcal P_N,\ d\le D}|R_N(d)|\). For the explicit sieve functions put

The implemented density theorem yields, for each \(\delta {\gt}0\), a fixed \(\epsilon {\gt}0\) and eventually for even \(N\)

The density input is supplied by the constructed Jurkat–Richert delay functions through the modern Suzuki varying-family comparison. Continuity at \(s=5\) permits a fixed small loss \(s=5-10\epsilon \). Ordinary Bombieri–Vinogradov pays \(\mathcal R_0\) at this power-reduced level; the removed exceptional primes receive a separate power-saving bound. Mertens normalization supplies the lower estimate \(XV_N\ge (20e^{-\gamma }-\delta )\mathcal X_N\) for any fixed positive margin. Combining the three eventual bounds gives, for every \(\eta {\gt}0\),

The proof chooses the density loss first, then invokes distribution at that same \(\epsilon \); all thresholds are intersected before choosing \(N\).

3.1.3 The varying-level upper penalty and its distribution bill

For each \(q\in \mathcal Q(N)\), condition the same complement carrier by \(q\mid N-p\). Its nominal mass is \(X/(q-1)\), including the lane \(q\mid N\); the exact discrepancy, rather than a reduced-residue approximation, handles that lane. The upper sieve uses

Integer support is \(d{\lt}\lfloor D_q\rfloor +1\). Summing the finite upper Rosser inequalities gives \(H\le \mathcal M_q+\mathcal R_q\), with \(\mathcal M_q\) the sum of \(X/(q-1)\) times the actual upper density sums, and \(\mathcal R_q\) the sum of their absolute-coefficient remainder bills.

The modern all-depth Suzuki theorem controls these density sums uniformly in the conditioned sieve. In the range used here its continuous factor is

Thus the density model is \(XV_N\sum _{q\in \mathcal Q(N)}F(s_q)/(q-1)\). Partial summation retains both \(s_q\) and the exact \(q-1\). For any margin, one chooses \(0{\lt}\epsilon {\lt}1/60\) and then obtains the coefficient \(B_1=8(\log 8+K/2)\), using the integrals defined above. Distribution is needed for the combined modulus \(qd\). On \(q\nmid N\) the finite bill is dominated by

where the fixed-endpoint error and the foundations’ prefix error are

Here \(\pi (N;m,l)\) counts primes at most \(N\) in residue class \(l\) modulo \(m\). Modulo one the canonical residue is zero; for modulus zero both maxima are defined to be zero. Thus this is the same genuine-\(L_*\) normalization as ordinary BV, but only the endpoint maximum is consumed in the displayed bill. The definitions and endpoint-to-prefix comparison are in ordinary AP errors; the exact weighted bill is in combined-modulus error sum. The map \((q,d)\mapsto qd\) is injective here: \(q\) is above every prime factor of \(d\). Finite Cauchy–Schwarz pays the weight with a \(9^{\omega }/d\) moment; ordinary BV is requested with saving \(2A+10\) to leave saving \(A\). Nonreduced lanes and exceptional support corrections are bounded separately and then absorbed into \(\mathcal X_N\).

The upper asymptotic allocates one third of its margin to density comparison, one third to prime partial summation, and one third to distribution. It concludes \(H(N)\le (B_1+\eta )\mathcal X_N\) eventually for even \(N\).

Using a base margin \(\eta /2\) and an upper margin \(\eta \) now gives

The last comparison uses the proved \(J-K/4\ge -0.0164725\) and a certified lower bound for \(\log 2\); it is an integral inequality, not floating-point quadrature.

3.1.4 Paying for rounded endpoints and repeated prime factors

Write \(W_{\rm dist}=|\mathcal C|-\frac12\sum _{p\in \mathcal C} \# \{ r\in [z,y):r\text{ prime},\ r\mid N-p\} \). For even \(N{\gt}2^{110}\), the implemented source comparison is \(W_{\rm src}-10N^{9/10}\le W_{\rm dist}\). It splits the two carriers into their intersection and exceptional fibres: the lower-floor fibre, \(p=2\), and the unit complement \(N-p=1\). Signed weights need not be monotone under enlarging the candidate set, so this comparison is necessary. Repeated medium factors have their own finite estimate:

Divisibility by \(r^2\) is bounded by counting quotients, giving at most \(6N^{9/10}\) candidate–prime pairs; each relevant valuation is at most ten.

Hence \(W_{\rm dist}-30N^{9/10}\le W\). Since \(60N^{9/10}\le \rho \mathcal X_N\) eventually for every \(\rho {\gt}0\), the source-to-corrected loss and the valuation loss fit within an arbitrary coefficient margin. Taking the source margin and the power-error margin to be \(\eta /2\) proves \(W(N)\ge (2.6408-\eta )\mathcal X_N\). This lower bound has already paid the entire prime-power penalty \(P\).

3.1.5 Moving the remaining triple penalty into a Selberg square

Only \(T\) remains to be subtracted. Put \(z_0=\lfloor N^{1/10}\rfloor \), \(y_0=\lfloor N^{1/3}\rfloor \), and let \(\mathcal L\) consist of prime pairs \((r,s)\) satisfying \(z_0{\lt}r\le y_0{\lt}s\) and \(rs^2\le N\). Liu’s characteristic weight \(b(a)\) is one exactly when \(a=rs\) for such a pair; it is unique, and sums involving \(b\) run over its support. On \(z\le r{\lt}y\le s\), let \(T_{\rm src}\) count \((r,s,u)\) with \((r,s)\in \mathcal L\), \(u\) prime, \(s\le u\), \(rsu\le N\), and \(N-rsu\in \mathcal C(N)\). A first-factor witness counted by \(T\) chooses one ordered pair \((s,u)\). The map to \((r,s,u)\) recovers the candidate as \(p=N-rsu\), so it is injective. Its only cutoff exceptions are \(r=z_0\) and \(s=y_0\); quotient counting pays both, giving \(T\le T_{\rm src}+13N^{9/10}\) once \(z_0\ge 2\).

For even \(N\) and \(R=\lfloor N^{1/4-\epsilon /2}\rfloor \ge 1\), put \(Q=\prod _{r\le R,\ r\ \text{prime},\ r\nmid N}r\) and \(\mathcal D=\{ d\mid Q:d\le R\} \). The Selberg square is

The optimal coefficients satisfy \(\lambda _1=1\), \(|\lambda _d|\le 1\), and vanish off \(\mathcal D\). If the candidate prime \(p=N-rsu\) does not divide \(Q\), its divisor packet contains only \(d=1\) and its square is one. Primes \(p\mid Q\) contribute to a residual \(E_Q\). Nonnegativity allows us to drop \(s\le u\) and enlarge the common pair region, yielding \(T_{\rm src}\le \mathcal B_\lambda +E_Q\).

For each source pair the exceptional residual injects into the prime divisors of \(Q\). Pair support has size at most \(\lceil N^{2/3}\rceil +1\), whereas \(Q\) has at most \(R+1\) prime divisors. Thus \(E_Q\le 6N^{11/12}\) for \(N\ge 1\) and \(\epsilon \ge 0\). Both this loss and the endpoint loss are smaller than \(C_A N/(\log N)^A\) for every fixed \(A{\gt}0\).

3.1.6 The square’s main term and the switched distribution error

Expanding the square replaces simultaneous divisibility by \([d_1,d_2]\mid N-au\). Finite interchange gives exactly

The error uses the same coefficients and the actual progression discrepancy \(\pi (N;a,d,l)-\operatorname {li}_2(N/a)/\varphi (d)\), where \(\pi (N;a,d,l)\) counts primes \(u\) with \(au\le N\) and \(au\equiv l\pmod d\).

The optimizer uses reverse-divisor Möbius inversion of the diagonal coordinates \(\mu (\ell )g(\ell )/\mathcal G\), where \(g(\ell )=\prod _{\substack {r\mid \ell \\ r\ \text{prime}}}(r-2)^{-1}\) and \(\mathcal G=\sum _{\ell \in \mathcal D}g(\ell )\). Diagonalizing the quadratic form proves its value is exactly \(1/\mathcal G\). The denominator estimate bounds this by \((8+\delta )\mathfrak S_{\mathrm{Liu}}(N)/\log N\) whenever \(\delta {\gt}0\) and \(0{\lt}\epsilon {\lt}\delta /(2(8+\delta ))\), eventually along even \(N\). The other factor must be estimated with the genuine logarithmic integral, not identified with its proxy \(x/\log x\).

Write the exact prime-pair reciprocal-log mass as

In logarithmic coordinates \(\alpha =\log r/\log N\) and \(\beta =\log s/\log N\), the limiting domain is

The sloping edge is precisely the condition \(rs^2\le N\); the finite sum still uses the rounded cutoffs defining \(\mathcal L\). The limiting prime reciprocal measures give the factors \(d\alpha /\alpha \) and \(d\beta /\beta \), while \(\log (N/(rs))=(1-\alpha -\beta )\log N\). The source integral is therefore

All denominators are positive on this domain. Partial fractions evaluate the inner integral as \(\log (2-3\alpha )/(1-\alpha )\), proving the reduction. The strict constant bound is proved for this integral itself: after the change \(t=(1-3\alpha )/(3-3\alpha )\) it becomes

On this interval the source bounds the logarithm above by

This follows from the logarithmic series with its bounded positive tail. Exact integration of \(3P_{\log }(t)/(1-3t)\), followed by a rational upper bound for \(\log (9/2)\), gives the strict inequality above. This is not a comparison of two preassigned decimals or a floating-point quadrature; see source integral and strict bound.

For each fixed \(\tau {\gt}0\), the actual transfer proves

It first fixes a fine logarithmic grid, then chooses the finite-\(N\) threshold for that grid. Choosing \(\tau \) inside the strict gap \(0.49254-I_{\rm Liu}\) gives \(S_{\rm rec}(N)\le 0.49254/\log N\) eventually, with the exact finite pair carrier unchanged. These are the actual producers in prime-pair transfer.

To pass to the genuine main mass, fix \(\kappa \ge 0\) and put

The equality is exact reindexing by the unique prime pair. Integration by parts gives \(|\Delta _{\kappa }(x)|\le C_{\Delta ,\kappa }x/(\log x)^2\) eventually. For every source pair, \(N/(rs)\ge N^{1/3}\), hence \(\log (N/(rs))\ge (\log N)/3\). Mertens bounds the reciprocal pair mass by a fixed constant \(B_{\rm pair}\), so uniformly over the pair set,

Thus for every fixed \(\kappa \ge 0\) and \(\eta {\gt}0\), eventually

The exact decomposition and summed correction are proved in genuine-li correction; unconditional genuine-li mass combines this correction with the proved reciprocal-log bound. The actual even-filter optimizer consumes that result in optimized main-term assembly. It fixes \(\delta =\eta =10^{-7}\) and \(0{\lt}\epsilon \le \epsilon _0=10^{-10}\); the coefficient product satisfies \((8+\delta )(0.49254+\eta )\le 3.94033\). Consequently the displayed main term \(M_1=H_2(N)/\mathcal G\) satisfies \(M_1\le 3.94033\mathcal X_N\) with a genuine, fixed coefficient margin.

The remaining distribution theorem concerns the convolution \(b(a)\) with primes. For \(A_1=\lfloor (\log N)^{2B}\rfloor +1\) and \(A_2=\lfloor N^{2/3}\rfloor \), define

The absolute value is outside the \(a\)-sum. Modulus one uses residue zero; modulus zero contributes zero. At the fixed normalization \(\kappa =2/\log 2\), the proved source theorem says that every \(\sigma {\gt}0\) admits \(C,B,N_0\) such that \(\sum _{q{\lt}N^{1/2}/(\log N)^B}E_q(\kappa )\le CN/(\log N)^\sigma \) for every \(N\ge N_0\). Its proof splits nonprincipal primitive conductors into low and high ranges, uniformly in the changing source and cofactor, and pays the principal character separately. Two extra logarithms pay the reciprocal-totient cofactor sum before this unweighted bound is obtained. See the source-linked statement in result 9. The Selberg expansion needs a weighted modulus sum. Finite Cauchy gives

Requesting \(\sigma =2A+11\) therefore leaves any prescribed saving \(A\).

Source support identifies the interval sum with the full supported sum. Replacing the strict source modulus range by the smaller closed range costs \(B\mapsto B+1\); changing the additive normalization to \(\kappa =2\) is paid separately. This supplies the canonical coprime distribution bound. In the square, \([d_1,d_2]\le R^2\le N^{1/2-\epsilon }\); regrouping coefficient pairs gives the \(3^{\omega (d)}\) majorant. The noncoprime \(a\)-part has an independent \(O(N^{9/10}(\log N)^2)\) bound, so it too fits every fixed inverse-log budget. The resulting estimate is \(|\mathcal E_\lambda |\le C N/(\log N)^A\), uniformly for the eventual admissible coefficient family, with \(C\) fixed before \(N\).

3.1.7 Closing one budget on the original representation count

Combine the square bound with the two finite corrections to obtain

eventually for even \(N\). In the actual assembly \(\epsilon =\epsilon _0\) and \(A=3\) are fixed first; the resulting constant is the sum of the square, endpoint, and paper-modulus residual constants.

Now choose the weighted-lower margin \(\eta =1/10000\). The uniform bound \(\mathfrak S_{\mathrm{Liu}}(N)\ge U{\gt}0\) makes the displayed remainder at most \(\mathcal X_N/10000\) eventually. Intersect these two events with the triple-bound event and \(N\ge 9\). The finite bridge then gives

This is the public count introduced at the start, with the same prime, square, and unit conventions.

Since \(\mathcal X_N{\gt}0\) for \(N{\gt}1\), a positive finite cardinality supplies a prime \(p\in G(N)\) and hence \(N=p+(N-p)\). The implemented existence route uses the same finite positivity lemma with the more generous margins \(1/10\), then extracts an eventual threshold \(N_0\) and expands the almost-prime predicate. Its conclusion permits either a prime partner or \(q=rs\) with \(r,s\) prime, including \(r=s\).