An incremental approximation to the prime-counting function via polygonal numbers
Abstract
We construct an incremental prime-counting approximation from corrected right-endpoint sums on polygonal-number partitions. For each fixed polygonal family, retaining completed contributions gives 𝒪(1) numerical operations per consecutive node and 𝒪(√X) operations for the full traversal through X, in the unit-cost model. The scheme uses a fixed number of working numerical words rather than repeated summation. Asymptotic formulas for the discretization error and terminal scale yield an unconditional relation with the classical normalized prime-counting error. We prove that the logarithmic mean of the normalized estimator error converges to 1/√2 if and only if the Riemann hypothesis holds, and give an independent Mellin proof of the converse. Under the hypothesis, this constant uniquely removes the limiting signed mean bias. Computations at powers of ten through 1019 with coefficient $0.7071$ show accuracy comparable to the Riemann approximation at the sampled arguments
Related
- On the counting function of square-full numbers
- Rational Points on Danielewski Surfaces over Finite fields and Polygonal Numbers
- Retriangular numbers and affine maps between polygonal numbers
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- Random field approximation and local counting statistics for the weakly interacting thermal Bose gas
BibTeX
@misc{samojluk2026incremental,
title = {{An incremental approximation to the prime-counting function via polygonal numbers}},
author = {Artur Samojluk and Artur Siemaszko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.31761},
eprint = {2606.31761},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.31761},
}
RIS
TY - UNPB
TI - An incremental approximation to the prime-counting function via polygonal numbers
AU - Artur Samojluk
AU - Artur Siemaszko
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2606.31761
UR - https://arxiv.org/abs/2606.31761
AB - We construct an incremental prime-counting approximation from corrected right-endpoint sums on polygonal-number partitions. For each fixed polygonal family, retaining completed contributions gives 𝒪(1) numerical operations per consecutive node and 𝒪(√X) operations for the full traversal through X, in the unit-cost model. The scheme uses a fixed number of working numerical words rather than repeated summation. Asymptotic formulas for the discretization error and terminal scale yield an unconditional relation with the classical normalized prime-counting error. We prove that the logarithmic mean of the normalized estimator error converges to 1/√2 if and only if the Riemann hypothesis holds, and give an independent Mellin proof of the converse. Under the hypothesis, this constant uniquely removes the limiting signed mean bias. Computations at powers of ten through 10^19 with coefficient $0.7071$ show accuracy comparable to the Riemann approximation at the sampled arguments
ER -