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On the Distribution of Prime Squares and Its Applications
Abstract
We prove that, for every A>0, all but OA(X(log X)−A) integers n<X satisfying n≡11pmod24 admit a representation n=a2+b2+p2, where p and a2+b2+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.
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BibTeX
@misc{mousavi2026distribution,
title = {{On the Distribution of Prime Squares and Its Applications}},
author = {Hamed Mousavi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00485},
eprint = {2611.00485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00485},
}
RIS
TY - UNPB
TI - On the Distribution of Prime Squares and Its Applications
AU - Hamed Mousavi
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00485
UR - https://arxiv.org/abs/2611.00485
AB - We prove that, for every A>0, all but O_A(X(log X)^−A) integers n<X satisfying n≡11pmod24 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.
ER -