Vol. 89, No. 1, 1980

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A double inversion formula

John David Brillhart

Vol. 89 (1980), No. 1, 7
Abstract

Let G be an abelian group and suppose {an} and {bn}, n 1, are sequences in G. Let p be an odd prime and set ηe = (e1∕p), the Legendre symbol, where e = pse1, s 0, pe1. Also, let χe± = (1 ± ηe)2. Define the sequence {cn} and {dn}, n 1, by

     ∑
cn =    (χ+e af + χ−e bf)
ef=n
(1)

and

     ∑
dn =    (χ−e af + χ+e bf).
ef=n
(2)

Theorem. For n 1 and μ the Möbius function,

     ∑       +      −
an =    μ(e)(χe cf + χe df)
ef=n
(3)

and

     ∑        −      +
bn =    μ (e)(χe cf + χ e df).
ef=n
(4)

Mathematical Subject Classification
Primary: 10A20, 10A20
Milestones
Received: 8 June 1979
Published: 1 July 1980
Authors
John David Brillhart