Re: sums related to fractional parts

David Bernier <david250@xxxxxxxxxxxx> wrote:
For a real number x, we have floor(x), which is the largest integer less
than or equal to x.

The fractional part of x, frac(x), can then be defined as:
frac(x) := x - floor(x).

So 0 <= frac(x) < 1. A related function is obtained by subtracting 1/2
from frac(x).
We have : -1/2 <= frac(x)-1/2 < 1/2 . Now we can define the
sawtooth function, or "saw" by
saw(x) := frac(x) - 1/2 if x is not in Z and
saw(x) := 0 if x is in Z.

Then we have -1/2 < saw(x) < 1/2 for all x in R.

Not that it has any pertinence to your question, but...

Using the floor function, which I'll denote using square brackets, your saw
function can be defined without having to define it separately for x in Z:

saw(x) := x + ([-x] - [x])/2 for all x in R

The series I have been looking at is:
saw(pi) + saw(2pi) + ... + saw(k*pi) , for various k.

Caesar: came(pi), saw(pi), ate(pi)

Cheers,
David

I drew a graph of the partial sums for k from 1 to 100,000.
This has a resemblance to the graph of |sin(x)| for 0<=x<= 3pi .

In fact, what I get is that the minimum of the sum for k from 1 to 100,
000 is for k = 66426, where the sum takes the value of ~= -0.88.

The graph was done using the C language and MatLab, and a check on the
value of the sum for k=66426 can be obtained with PARI-gp:

(07:27) gp > sum(X=1,66426, X*Pi - floor(X*Pi) - 1/2)

%1 = -0.88209459147...

Contrary to my expectation at first, the plot of the partial sums
doesn't look random at all.
I cannot explain these observations and I don't know the behavior of the
partial sums
for large k.

David Bernier
.

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