On Representations of Even Integers as a Sum of Two Semiprimes

1. Introduction

Let $S_2=\{n\ge 2:n=pq\mathrm{with}p\le q\mathrm{primes}\}$. Consider the weighted representation function

$$R\left(N\right):=\sum _{\begin{array}{c}{s}_1+s_2=N\\ s_1,s_2\in S_2\end{array}}W\left(s_1\right)W\left(s_2\right),$$

(1)

where W is a nonnegative smooth weight localizing $s_i\asymp N$. The main result is an asymptotic of order N with a power saving in $logN$. The method uses the major/minor arc decomposition and a bilinear Bombieri–Vinogradov type estimate for semiprimes.

2. Statement of Results

Theorem 1. 

Let W be a smooth nonnegative bilinear weight supported on $uv\asymp N$ with $u\le v$, arising from a finite smooth dyadic partition of unity, and set $R\left(N\right)$ as in (1). There exist absolute constants $\delta >0$ and a multiplicative singular series $\mathfrak{S}\left(N\right)={\prod }_p{\sigma }_p\left(N\right)$ such that, for all sufficiently large even N,

$$R\left(N\right)=\mathfrak{S}\left(N\right)N+O\left({\displaystyle \frac{N}{{(logN)}^{1+\delta }}}\right).$$

Moreover, there exist constants $0<c_1\le c_2<\infty$ with $c_1\le \mathfrak{S}\left(N\right)\le c_2$ for all even N.

Corollary 1 

(Semiprime Goldbach for large even integers). For all sufficiently large even integers N, there exist semiprimes $s_1,s_2\in S_2$ with $s_1+s_2=N$.

Proof. 

Choose W to majorize the indicator on a fixed positive-proportion window (e.g. $W\ge 1$ on $[N/4,3N/4]$ and supported on $x\asymp N$). By Theorem 1, $R\left(N\right)=\mathfrak{S}\left(N\right)N+O\left(N{(logN)}^{-1-\delta }\right)$ with $\mathfrak{S}\left(N\right)$ bounded below by a positive constant along even N; hence $R\left(N\right)>0$ for N large, which forces the existence of a representation.    □

3. Circle Method Outline

Let $S\left(\alpha \right)={\sum }_{s\in S_2}W\left(s\right){\mathrm{e}}^{2\pi i\alpha s}$. Then

$$R\left(N\right)={\int }_0^{1}S{\left(\alpha \right)}^2{\mathrm{e}}^{-2\pi i\alpha N}d\alpha .$$

Major arcs are handled by a local main term plus an error; minor arcs are bounded via Gallagher’s lemma and a multiplicative large sieve adapted to the semiprime model. The required distribution up to $Q\le N^{\theta -\epsilon }$ with $\theta >1/2$ is supplied by the bilinear estimate below.

4. A Bilinear Bombieri–Vinogradov Input

Let X be large and $U,V$ be dyadic with $UV\asymp X$, $U\le V$. Let $\varpi (u,v)$ be a smooth nonnegative cutoff supported on $u\asymp U$, $v\asymp V$, with derivatives bounded up to a fixed order, and let $a_u,b_v$ be bounded coefficients modeling prime-like weights.

Proposition 1. 

There exists $\theta >1/2$ and $\eta >0$ such that, uniformly for $Q\le X^{\theta -\epsilon }$,

$$\sum _{q\le Q}\underset{(a,q)=1}{max}\left|\sum _{\begin{array}{c}u,v\ge 1\\ uv\equiv a\left(\mathrm{mod}q\right)\end{array}}\varpi (u,v)a_u{b}_v-{\displaystyle \frac{1}{\phi \left(q\right)}}\sum _{\begin{array}{c}u,v\ge 1\\ (uv,q)=1\end{array}}\varpi (u,v)a_u{b}_v\right|{\ll }_{\epsilon ,\varpi }X{(logX)}^{-1-\eta }.$$

Remark 1. 

The estimate follows from the dispersion method and the multiplicative large sieve in standard bilinear forms.

5. Finite Verification (Computational Note)

A numerical check (unweighted) indicates that the only even integers $N\le {10}^6$ that are not a sum of two semiprimes are $2,4,6,22$. In particular, every even $N\ge 24$ up to ${10}^6$ admits a representation, consistent with Theorem 1. A standalone Python script is provided in Appendix A to reproduce and extend this verification.