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Ramanujan and Eckford Cohen totients from Visible Point Identities

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.