A prime plus a square: A conditional 11/20 upper bound
Abstract
Let E□(N) count the positive integers at most N that cannot be expressed as a prime plus a nonnegative integer square. Assuming that every nontrivial zero of every primitive Dirichlet L-function has real part at most 7/8, we prove E□(N)≪eps N11/20+eps for every eps>0. The proof combines a conductor-weighted mean-square estimate with a uniform estimate for certain families having a prescribed quadratic component. For these families, the large sieve depends on the varying part of the conductor, while an approximate functional equation controls the fourth moment using the full conductor. A zero-detection argument then absorbs all bands of zeros with real part at least 129/167. The remaining bands and the minor arcs give the exponent 11/20. The proof includes even denominators, completion and positivity of the singular series, and removal of proper prime powers.
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BibTeX
@misc{pintz2026prime,
title = {{A prime plus a square: A conditional 11/20 upper bound}},
author = {János Pintz and Ingo Althöfer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21655},
eprint = {2610.21655},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21655},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A prime plus a square: A conditional 11/20 upper bound
AU - János Pintz
AU - Ingo Althöfer
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21655
UR - https://arxiv.org/abs/2610.21655
AB - Let E_□(N) count the positive integers at most N that cannot be expressed as a prime plus a nonnegative integer square. Assuming that every nontrivial zero of every primitive Dirichlet L-function has real part at most 7/8, we prove E_□(N)≪_eps N^11/20+eps for every eps>0. The proof combines a conductor-weighted mean-square estimate with a uniform estimate for certain families having a prescribed quadratic component. For these families, the large sieve depends on the varying part of the conductor, while an approximate functional equation controls the fourth moment using the full conductor. A zero-detection argument then absorbs all bands of zeros with real part at least 129/167. The remaining bands and the minor arcs give the exponent 11/20. The proof includes even denominators, completion and positivity of the singular series, and removal of proper prime powers.
ER -