A generalized Mobius transform, arithmetic Fourier transforms, and primitive roots

A general approach to arithmetic Fourier transforms is developed. The implementation is based on sine and cosine killer procedures pertaining to a generalized Möbius transform involving reduced periodic multiplicative arithmetical functions. It is shown that cosine killer procedures exist whenever one half of Euler’s totient function of the order of the transform is odd. Primitive roots and indices with respect to primitive roots play an important part in the derivation of the results.