[seqfan] Re: Sum over tangents

Richard J. Mathar mathar at mpia-hd.mpg.de
Fri Apr 26 14:09:00 CEST 2013


http://list.seqfan.eu/pipermail/seqfan/2013-April/011083.html says

cgr> A086227 is defined as "a(n)=1/(4i)*sum( i^k*tan(k*Pi/4/n)) where 1<=k<=4n
cgr> and (k,n)=1" with no further comment. But it seems that the absolute value
cgr> of this sequence gives the class number for Q(sqrt(-4*n^2)). Is this true?
cgr> What is the connection between these definitions?

These might be related:

J. H. Conway, C. Radin, L. Sadun,
On angles whose squared trigonometric functions are rational
Discr. Comput. Geom. 22 (1999) 321-332
doi:10.1007/PL00009463
ftp://ftp.ma.utexas.edu/pub/papers/radin/geodetic.ps

See Corollary 5.2 in
B. C. Berndt, A. Zaharescu,
Finite trigonometric sums and class numbers
doi: 10.1007/s00208-004-0559-5
Math. Ann. 330 (2004) 551-575
http://www.math.uiuc.edu/~berndt/articles/trigtheta3.pdf

and perhaps
http://www.oberlin.edu/faculty/jcalcut/tanpap.pdf


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