We establish pointwise almost everywhere convergence for the polynomial multiple ergodic averages 1N ₍=₁N (n) f₁ (T^P₁ (n) x) fₖ (T^Pₖ (n) x) as N, where is the von Mangoldt function, T X X is an invertible measure-preserving transformation of a probability space (X, ), P₁, , Pₖ are polynomials with integer coefficients and distinct degrees, and f₁, , fₖ L^ (X). This pointwise almost everywhere convergence result can be seen as a refinement of the norm convergence result obtained in Wooley--Ziegler (Amer. J. Math, 2012) in the case of polynomials with distinct degrees. Building on the foundational work of Krause--Mirek--Tao (Ann. of Math. , 2022), Kosz--Mirek--Peluse--Wright (arXiv: 2411. 09478, 2024), and Krause--Mousavi--Tao--Ter\"av\"ainen (arXiv: 2409. 10510, 2024), we develop a multilinear circle method for von Mangoldt-weighted (equivalently, prime-weighted) averages. This method combines harmonic analysis techniques across multiple groups with the newest inverse theorem from additive combinatorics. In particular, the principal innovations of this framework include: (i) an inverse theorem and a Weyl-type inequality for multilinear Cram\'er-weighted averages; (ii) a multilinear Rademacher-Menshov inequality; and (iii) an arithmetic multilinear estimate.
Renhui Wan (Wed,) studied this question.