Resumen de On certain trigonometric sums in several variables

R. Sivaramakrishnan, Pentti Haukkanen

• The Hilbert space structure of totally even functions (mod $r$) which depend on extended Ramanujan sums is described. The function $\varepsilon_k$ defined as the quotient of Jordan's $J_k$-function and Euler's $\phi$-function is introduced as a new generalization of the Dedekind $\psi$-function. Using the basic methods of totally even functions (mod $r$), we point out that $\varepsilon_k$ has also a purpose to serve in obtaining the $k$-dimensional analogue of an identity due to P. Kesava Menon.