[seqfan] Re: identity

Richard Mathar mathar at strw.leidenuniv.nl
Thu Feb 4 14:49:58 CET 2010

On the observation
sum( phi(2*k+1)*round((n-1)/(2*(2*k+1))), k = 0..(n/2)-1 ) = n^2/4
in http://list.seqfan.eu/pipermail/seqfan/2010-February/003583.html 
(and its obvious generalization to odd n if the r.h.s is replaced by [n/2]^2 ):

This looks vaguely like an appication of (1) in Mercier's article
"Sum containing the fractional parts of numbers"
Rocy Mountain J. of Math, 15 (2) (1985) 513

sum_{k=0.. n/2-1} phi(2k+1)* [ (n+2k-2)/(2(2k+1))]
sum_{k=1,3,5,7.. n-1} phi(k)* [ (n-3+k)/(2k)]
appears to be of the form on his right hand side.

Another format of proof might observe that the sum is underneath a parabolic
section of a curve in the (k,n) coordinates. So writing phi(k) by mobieus
inversion might allow resummation over the divisors of n..
(see P Haukkanen in Int J Math Math Sci 19 (2) (1996) 209-218 ,
 for a useful collection of transformations..)

Richard Mathar

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