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On the Fourier transform of the greatest common divisor

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journal contribution
posted on 29.03.2022, 06:14 authored by Pieter Van Der KampPieter Van Der Kamp
The discrete Fourier transform of the greatest common divisor (formula presented) with αm a primitive m-th root of unity, is a multiplicative function that generalizes 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, id[a] = id * c•(a), and on the other hand it can be written as a generalized convolution product, id[a] = id *a φ. We show that id[a](m) counts the number of elements in the set of ordered pairs (i, j) such that i · j ≡ a mod m. Furthermore we generalize 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.

Funding

This research has been funded by the Australian Research Council through the Centre of Excellence for Mathematics and Statistics of Complex Systems.

History

Publication Date

01/01/2013

Journal

Integers

Volume

2013

Article Number

A24

Pagination

16p.

Publisher

Colgate University, Charles University, and DIMATIA

ISSN

1867-0652

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All works of this journal are licensed under a Creative Commons Attribution 4.0 International License so that all content is freely available without charge to the users or their institutions. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles in this journal without asking prior permission from the publisher or the author. This is in accordance with the BOAI definition of open access.

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