The Ramanujan Journal

, Volume 35, Issue 2, pp 205–241 | Cite as

Ramanujan sums as supercharacters

  • Christopher F. Fowler
  • Stephan Ramon Garcia
  • Gizem Karaali
Article

Abstract

The theory of supercharacters, recently developed by Diaconis-Isaacs and André, is used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and provides many novel formulas. In addition to exhibiting a new application of supercharacter theory, this article also serves as a blueprint for future work since some of the abstract results we develop are applicable in much greater generality.

Keywords

Ramanujan sum Multiplicative function Arithmetic function Even function modulo n Supercharacter theory Representation Supercharacter Kronecker product 

Mathematics Subject Classification

11L03, 11A25 20C15 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Christopher F. Fowler
    • 1
  • Stephan Ramon Garcia
    • 2
  • Gizem Karaali
    • 2
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of MathematicsPomona CollegeClaremontUSA

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