Ramanujan Sums

  • Paul J. McCarthy
Part of the Universitext book series (UTX)


If a and b are integers let
$$ e(a,b) = {e^{{\frac{{2\pi ia}}{b}}}} $$
Let n be an integer, positive, negative or zero, and let r be a positive integer. Consider the sum
$$ c(n,r) = \sum\limits_{{(x,r) = 1}} {e(nx,r)} $$
Usually, the sum is taken over all × such that 1 ≤ x ≤ r and (x, r) = 1, but it could be over any reduced residue system (mod r). This is because, if x ≡ x′ (mod r) then e(nx, r) = e(nx′, r). The sum c(n,r) is called a Ramanujan sum. For fixed r, and with n restricted to the positive integers, we obtain an arithmetical function c(·, r). Some authors devote this function by cr, so that cr(n) = c(n, r)


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Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Paul J. McCarthy
    • 1
  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA

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