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The sinc function is defined as:
Prove that for any couple of real numbers in the following result holds:
due to the addition formulas for the sine and cosine functions it is enough to prove the equality
which is an immediate consequence of the Fourier series by integrating the sawtooth wave function;
Let be a real number. Evaluate the series:
Since the summands are all positive , we can sum by triangles. Thus,
where is the Riemann zeta function.
Let denote the floor function. Evaluate the series
First of all we note that and are never squares. Thus, there exists a positive integer such that
It is easy to see that and thus we conclude that
Now is equal to the even number if-f
Hence, since the series is absolutely convergent we can rearrange the terms and by noting that the finite sums are telescopic , we get that:
Compute the series
Successively we have: