The discrete Fourier transform of $r$-even functions

Abstract

We give a detailed study of the discrete Fourier transform (DFT) of $r$-even arithmetic functions, which form a subspace of the space of $r$-periodic arithmetic functions. We consider the DFT of sequences of $r$-even functions, their mean values and Dirichlet series. Our results generalize properties of the Ramanujan sum. We show that some known properties of $r$-even functions and of the Ramanujan sum can be obtained in a simple manner via the DFT.

15 pages

References in corpus (2)

• Discrete Ramanujan-Fourier Transform of Even Functions (mod $r$)
• An analogue of Ramanujan's sum with respect to regular integers (mod $r$)

Cited by in corpus (7)

• Restricted linear congruences
• Proofs, generalizations and analogs of Menon's identity: a survey
• Dragging the roots of a polynomial to the unit circle
• The Discrete Fourier Transform of $(\mathbf{r},\mathbf{s})-$even functions
• Arithmetical properties at the level of idempotence
• Saito duality and the classical theory of arithmetic functions
• Menon-type identities concerning Dirichlet characters