Closed-Form Orthogonal Ramanujan Integer Basis

In this letter, a closed-form orthogonal Ramanujan integer basis is proposed and obtained by performing Gram-Schmidt process from the Ramanujan sum and its circular shift. It has a surprisingly simple and sparse form, which is better than the original complete Ramanujan basis. The relationship between the original basis and proposed basis is also clearly illustrated. Therefore, the original basis can be replaced easily.