Re: sawtooth*cotangent sums (variation on Dedekind sums)

On May 26, 8:01 am, David Bernier <david...@xxxxxxxxxxxx> wrote:
David Bernier wrote:

[...]

I made a plot of q(a,1999)/1999 as a function of a, for
1<=a<=1998. I chose 1999 because it is a prime number.

As above,
q(a, n) := sum_{k=1 ... n-1} { ((k/n))*cotan(pi*a*k/n) } .

A correction:
If
t(a,n) := sum_{k=1 ... n-1} { ((a * k/n)) * cotan(pi*k/n) },
for 1<=a<=n-1,
then the graph and the examples really refer to
t(a,1999)/1999 as a function of a, not
q(a,1999)/1999 as defined previously.

For example, with PARI-gp, for a=666:
q(666,1999)/1999 =
sum(X=1,1998,(a*X/1999-floor(a*X/1999)-1/2)*cotan(Pi*X/1999))/1999
~= 0.49200413454

but with a = 667,
q(667,1999) ~= -0.546811974 .

Correction:

t(667,1999)/1999 ~= -0.546811974 .

and earlier:
t(666,1999)/1999 ~= 0.49200413454

Note that 1999/3 = 666 + 1/3 .

I uploaded the plot as a *.png file here:
<http://www.geocities.com/ezcos/saw1999.png>
[...]

I have a second graph ...

_Definition_:
t(a,n) := sum_{k=1 ... n-1} { ((a * k/n)) * cotan(pi*k/n) }
( for a in {1, ... n-1}. )

The graph for
a |-> t(a,1999)/1999 appears at
< http://www.geocities.com/ezcos/saw1999.png > and

the graph for
a |-> t(a,27689)/27689 appears at
< http://www.geocities.com/ezcos/saw27k.png >

There is a sharp drop in the graphs from
a = (n-1)/2 to a= (n+1)/2 .

The magnitude of the drops appears to increase
quite slowly with n.

Another sample value:
t(500001,1000003/1000003 ~= 1.8878
-----------------------------------------------------------------
This according to PARI-gp:

(01:52) gp > sum(X=1,n-1,(X*m/n-floor(X*m/n)-1/2)*cotan(Pi*X/n))/n
%14 = 1.8878....
(01:52) gp > m
%15 = 500001
[...]
(01:53) gp > n
%18 = 1000003
------------------------------------------------------------------

The graphs for n=1999 and n=27689 look quite similar to me.

David Bernier



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