This paper gives the first explicit quantitative bounds that measure how ‘‘random’’ the primes look inside short intervals. The authors study the von Mangoldt function, a standard arithmetic function that highlights prime numbers, and compare it to a carefully corrected model. They show that on intervals of length at least X^{5/8+ε} the difference has small Gowers uniformity norms. Small Gowers norms mean there are no large, otherwise-hidden algebraic patterns in the primes in those intervals, and the savings the authors obtain are quasipolynomial rather than merely qualitative.

At a high level, a Gowers norm is a way to test for structured patterns among numbers. If the norm is small then the sequence behaves like random noise for a wide class of patterns. The paper works with a version of the von Mangoldt function that is normalised and compared to a ‘‘Cramér’’ style model, further refined to account for a possible exceptional zero of certain L-functions (a so-called Siegel zero). That refined model is subtracted off so the authors can test the remaining part for uniformity.

To get explicit bounds the authors combine several deep tools and add a new ingredient. They start from a recent inverse theorem for Gowers norms and then analyse the resulting correlation with so-called nilsequences, objects that capture structured, algebraic behaviour. A main new contribution is an efficient non-abelian Type II inverse theorem in which the dependence on the nilsequence dimension is made explicit. Technically, they apply Leng’s efficient equidistribution result in four variables directly to an unweighted corner correlation. This replaces an earlier factorisation-and-subgroup step used in prior work and leads to better, trackable quantitative estimates. The proof also uses the standard Heath–Brown identity to split the von Mangoldt function into Type I/Type II pieces and then concatenates many local factorisations to get a global structural reduction.