ABSTRACT
A function f : ℤ × ℕ → ℂ is said to be a modular arithmetical function with modulus r, if f (n′, r) = f (n, r) whenever n′ ≡ n(mod r). Eckford Cohen’s class of even functions (mod r), namely, the class of functions f for which f (n, r) = f ((n, r), r) where (n, r) denotes the g.c.d of n and r, belongs to the class of modular functions with modulus r. The arithmetical representation of f as a finite Fourier series is obtained using an orthogonal property of Ramanujan sums https://www.w3.org/1998/Math/MathML"> C ( n , r ) = ∑ h ( mod r ) ( h , r ) = 1 exp ( 2 π i n h r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351023344/b6c68d68-1226-4347-aaa8-58dfe0c0d0a9/content/eq768.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , as observed by Eckford Cohen [ 4 ].
