On the Fourier transform of the greatest common divisor

2012/01/16 by Peter H. van der Kamp, van der Kamp, Peter H.
Mathematics · #05A15 #11A05 #11A07 #11A25 #40B05 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:05A15 #msc:11A05 #msc:11A07 #msc:11A25 #msc:40B05

paper · pdf · doi:10.48550/arxiv.1201.3139

3 figures, submitted to Proceedings of the American Mathematical Society

arxiv created 2012/01/16 · arxiv updated 2012/01/17

Abstract

The discrete Fourier transform of the greatest common divisor is a multiplicative function that generalises both the gcd-sum function and Euler's totient function. On the one hand it is the Dirichlet convolution of the identity with Ramanujan's sum, and on the other hand it can be written as a generalised convolution product of the identity with the totient function. We show that this arithmetic function of two integers (a,m) counts the number of elements in the set of ordered pairs (i,j) such that i*j is equivalent to a modulo m. Furthermore we generalise a dozen known identities for the totient function, to identities which involve the discrete Fourier transform of the greatest common divisor, including its partial sums, and its Lambert series.

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