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First page of On the Distribution of Prime Squares and Its Applications

On the Distribution of Prime Squares and Its Applications

Hamed Mousavi

math.NT Oct 31, 2026 · v1 math.CO math.DS
formalization number-theory analysis
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TL;DR
An accompanying Lean 4 formalization of the paper's results, generated with ChatGPT assistance, is publicly available but not yet independently reviewed.
Abstract
We prove that, for every \(A>0\), all but \(O_A(X(\log X)^{-A})\) integers \(n<X\) satisfying \(n\equiv11\pmod{24}\) admit a representation \[ n=a^2+b^2+p^2, \] where \(p\) and \(a^2+b^2+1\) are prime and \((a,b)=1\). For ergodic averages along prime squares, we prove a universal divergence result for integrable functions, while establishing almost-everywhere convergence for near-integrable functions. We also quantify how the bounds used to establish convergence behave as one approaches integrability. Moreover, we prove endpoint improving and sparse bounds for the corresponding discrete averages, with an explicit Orlicz loss.
Problem

The paper studies the sequence of prime squares from three angles. The first is an exceptional-set problem of Goldbach type for representations n=a^2+b^2+p^2. The second is pointwise ergodic theory for averages along prime squares at or near L^1. The third is improving and sparse inequalities for the corresponding discrete averages.

Approach

Circle-method estimates for quadratic exponential sums over primes are combined with Landau–Page treatment of exceptional characters and a high–low decomposition. This decomposition yields fixed-scale improving inequalities, and maximal ℓ^2 estimates plus a stopping-time iteration upgrade them to sparse bounds. LaVictoire's criterion, together with Poisson spacing of quadratic residues (Granville–Kurlberg), gives L^1 divergence. A Lean 4 development accompanying the paper was produced with ChatGPT assistance.

Results

For every A>0, all but O_A(X(log X)^{-A}) integers n<X with n≡11 (mod 24) are of the form a^2+b^2+p^2, with p and a^2+b^2+1 prime and (a,b)=1. Prime-square ergodic averages are persistently universally L^1-bad but converge almost everywhere for an explicit Orlicz class near L^1. Endpoint improving and sparse bounds are proved with an explicit Orlicz loss.