Vol. 89, No. 1, 1980

Recent Issues
Vol. 294: 1
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
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