Fractional-polynomial prime-pair correlation conjecture

Let $f(x)=a_1x^{c_1}+a_2x^{c_2}+\ldots+a_kx^{c_k}$, where $A>1$, $c_1>0$ is a non-integer, $a_1\ne0$, and $c_1>c_2>\ldots>c_k$. For $N$ sufficiently large, set

$$M=\frac{N^{1/c_1}}{\log^B N}.$$

Fractional-polynomial prime-pair conjecture. There exists $B=B(f,A)$ such that

$$\mathop{\mathbb{E}}_{m\leqslant M}\mathop{\mathbb{E}}_{n\leqslant N}\Lambda(n)\Lambda\bigl(n+\lfloor f(m)\rfloor\bigr)=1+O\left(\frac{1}{\log^A N}\right).$$

This extends the paper's result to all fractional polynomials, including the previously open range $1<c<2$. The proposed approach would require a Vinogradov mean value theorem for fractional polynomials; the authors also expect analogous results for suitable Hardy-field functions.