The Fourier Transform of Functions of the Greatest Common Divisor
نویسنده
چکیده
We study discrete Fourier transformations of functions of the greatest common divisor: n ∑ k=1 f((k, n)) · exp( − 2πikm/n). Euler’s totient function: φ(n) = n ∑ k=1 (k, n) · exp(−2πik/n) is an example. The greatest common divisor (k, n) = n ∑ m=1 exp(2πikm/n) · ∑ d|n cd(m) d is another result involving Ramanujan’s sum cd(m). The last equation, interestingly, can be evaluated for k in the complex domain.
منابع مشابه
Discrete Ramanujan-Fourier Transform of Even Functions (mod $r$)
An arithmetical function f is said to be even (mod r) if f (n) = f ((n, r)) for all n ∈ Z + , where (n, r) is the greatest common divisor of n and r. We adopt a linear algebraic approach to show that the Discrete Fourier Transform of an even function (mod r) can be written in terms of Ramanujan's sum and may thus be referred to as the Discrete Ramanujan-Fourier Transform.
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تاریخ انتشار 2008