Let us move on to the series version. We also claim that
We also have
This continues to hold until but fails when we include the term. Coincidence? We might hope, naïvely, that the correct result is
This is in fact true, for . Number theorists will be delighted to learn that this follows from the Poisson summation formula.
(Poisson summation formula)
Let be compactly supported, piecewise continuous and continuous at integer points. Then
The previous observation follows from taking , which satisfies the hypothesis of the theorem (it is in fact continuous for ). The Fourier inversion theorem then tells us . So the right-hand side is the sum in question, and is the Borwein integral. Our previous analysis shows that the support of is . So vanishes at non-negative integers whenever .
It is common for the theorem to be stated for Schwarz functions instead. However, our function is not smooth, but the same proof goes through under our hypothesis.
[Proof of theorem] Set
Then, . Note that the sum converges since is compactly supported, and is continuous at since is continuous at integer points. Of course, it is also piecewise continuous, since in each open neighbourhood, the sum is finite. So we know the Fourier series of converges at . Recall that the Fourier series is