Year: 2026
Type: preprint
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 $R_k$, 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.
Source: arXiv (Cornell University)
Author: Belton, Martin
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221072282