The probabilistic independence conjecture for Goldbach events

Counterexample: the conjecture fails uniformly for every sufficiently large odd $N$.

The exact statement is Conjecture 1.5 of Q. Ma and R. Zhang, arXiv:2411.06978v3, subsequently published in the Canadian Journal of Mathematics, doi:10.4153/S0008414X25101090. The source assumes $N$ is odd throughout, and the counterexample respects that restriction.

Let

$$\Omega_N=\{(a,b,c)\in\mathbb N^3:a+b+c=N\}, \qquad |\Omega_N|=\binom{N-1}{2}\sim\frac{N^2}{2}.$$

Write $A_N=\{a\text{ is prime}\}$ and $A'_N=\{b,c\text{ are prime}\}$. Exact counting gives

$$|A_N|=\sum_{\substack{p<N\p\text{ prime}}}(N-p-1), \qquad |A'_N|=\#\{(p,q):p,q\text{ prime},\ p+q<N\}.$$

The prime number theorem alone yields

$$|A_N|\sim\frac{N^2}{2\log N}, \qquad |A'_N|\sim\frac{N^2}{2\log^2N}.$$

Indeed, the measures

$$\nu_N=\frac{\log N}{N} \sum_{\substack{p\le N\p\text{ prime}}}\delta_{p/N}$$

converge weakly to Lebesgue measure on $[0,1]$; integrating $1-x$ and the indicator of $x+y<1$ gives the two constants $1/2$. Consequently

$$\Pr(A_N)\sim\frac1{\log N}, \qquad \Pr(A'_N)\sim\frac1{\log^2N}.$$

On the other hand, the source's own equation (1.1), Vinogradov's ternary Goldbach theorem, gives for odd $N$

$$|A_N\cap A'_N| =\frac{(1+o(1))\mathfrak G(N)N^2}{2\log^3N},$$

where

$$\mathfrak G(N)= \prod_{\ell\mid N}\left(1-\frac1{(\ell-1)^2}\right) \prod_{\ell\nmid N}\left(1+\frac1{(\ell-1)^3}\right),$$

with products over primes. Therefore the true asymptotic is

$$\boxed{\displaystyle \frac{\Pr(A_N\cap A'_N)} {\Pr(A_N)\Pr(A'_N)} =(1+o(1))\mathfrak G(N).}$$

Since $N$ is odd, the factor at $\ell=2$ equals $2$. Hence, for every odd $N$,

$$2C_2\le\mathfrak G(N)\le C_3,$$

where

$$C_2=\prod_{\ell\ge3\ \mathrm{prime}} \left(1-\frac1{(\ell-1)^2}\right)=0.6601618\ldots, \quad C_3=2\prod_{\ell\ge3\ \mathrm{prime}} \left(1+\frac1{(\ell-1)^3}\right)=2.3009615\ldots.$$

In particular,

$$2C_2\ge 2\left(1-\sum_{j\ge1}\frac1{(2j)^2}\right) =2-\frac{\pi^2}{12}>1.$$

Both endpoints are sharp: odd primorials give $\mathfrak G(N)\to2C_2$, while odd primes give $\mathfrak G(N)\to C_3$. Thus

$$\liminf_{\substack{N\to\infty\N\ \mathrm{odd}}} \frac{\Pr(A_N\cap A'_N)}{\Pr(A_N)\Pr(A'_N)} =2C_2>1, \qquad \limsup_{\substack{N\to\infty\N\ \mathrm{odd}}} \frac{\Pr(A_N\cap A'_N)}{\Pr(A_N)\Pr(A'_N)} =C_3.$$

The proposed asymptotic independence is therefore false throughout its intended odd-integer domain. No binary Goldbach assumption is needed.