Ramanujan and Eckford Cohen totients from Visible Point Identities

TL;DR

This paper extends Ramanujan's trigonometric function to multiple dimensions, linking it to visible point identities and lattice sums, and introduces new generating functions generalizing classical number-theoretic functions.

Contribution

It provides a multidimensional extension of Ramanujan and Eckford Cohen totients, connecting them to lattice point identities and deriving new generating functions.

Findings

Extended Ramanujan function to arbitrary dimensions

Linked to visible point vector identities and lattice sums

Derived new generating functions generalizing classical totients

Abstract

We define an extension of the Ramanujan trigonometric function to arbitrary dimensions, and give the Dirichlet series generating function. The extension was first given by Eckford Cohen long ago. This links directly to visible point vector identities, and possibly to lattice sums in Physics and Chemistry presented by Baake et al. New generating functions and summations are given here, generalizing the Ramanujan function, Euler totient and the Jordan totient functions, based on visible lattice point ideas.