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A note on Riemann's R-function and almost primes
Abstract
The real-axis jump of the prime-zeta function gives an integral form of Gram's series for Riemann's R-function. Applying the same operation to the classical almost-prime generating functions defines a smooth family Rk, with R1=R. We give the construction and a convergence argument, and explain why its expansion at every fixed logarithmic order is the Selberg--Delange expansion. The distinction lies in the smaller real-axis contributions retained by the definition, rather than in new asymptotic coefficients. A finite Perron decomposition also identifies the additional contour terms required for exact recovery of the count.
Related
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- Möbius function and primes: an identity factory with applications
BibTeX
@misc{belton2026note,
title = {{A note on Riemann's $R$-function and almost primes}},
author = {Martin Belton},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11024},
eprint = {2610.11024},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11024},
}
RIS
TY - UNPB
TI - A note on Riemann's R-function and almost primes
AU - Martin Belton
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11024
UR - https://arxiv.org/abs/2610.11024
AB - The real-axis jump of the prime-zeta function gives an integral form of Gram's series for Riemann's R-function. Applying the same operation to the classical almost-prime generating functions defines a smooth family R_k, with R_1=R. We give the construction and a convergence argument, and explain why its expansion at every fixed logarithmic order is the Selberg--Delange expansion. The distinction lies in the smaller real-axis contributions retained by the definition, rather than in new asymptotic coefficients. A finite Perron decomposition also identifies the additional contour terms required for exact recovery of the count.
ER -