Almost-identity: $\prod_{k=0}^N\text{sinc}\left(\frac{x}{2k+1}\right) = \frac{1}{2}$

Show that the identity
$$\int_0^\infty \prod_{k=0}^N \text{sinc}\left(\frac{x}{2k+1}\right)\,{\rm d}x – \sum_{n=1}^\infty \prod_{k=0}^N \text{sinc}\left(\frac{n}{2k+1}\right) = \frac{1}{2}$$ where $\text{sinc}(x) = \frac{\sin(x)}{x}$ holds for $N=0,1,2,\ldots,40000$ but fails for all larger $N$.

I remember seeing this strange identity a few years ago and it stuck to my mind, but unfortunately I can’t find the source of this right now and it’s reconstructed from memory (and checked with a computer for small $N$ although $40000$ might not be accurate). This is why I’m asking it here.

If I remember correctly it’s closely linked to Fourier transforms and for $N$ larger than $\sim 40000$ the difference between the left and right hand side should be smaller than $\sim 10^{-10000}$ so the agreement is extremely good.

Do anyone know the source of this problem or otherwise how to solve it? What is the theory behind it (i.e. why does it break down at some finite value)?

Solutions Collecting From Web of "Almost-identity: $\prod_{k=0}^N\text{sinc}\left(\frac{x}{2k+1}\right) = \frac{1}{2}$"

There is a theorem stated on page 2 of this paper, which states that for $N+1 > 1$ positive numbers $a_0,a_1,\ldots,a_N > 0$ the identity

$$\dfrac{1}{2}+\sum_{n = 1}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kn) = \int_{0}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kx)\,dx$$

holds provided that $\displaystyle\sum_{k = 0}^{N}a_k \le 2\pi$. (Also, for $N = 0$, the identity holds if $a_0 < 2\pi$).

Since $\displaystyle\sum_{k = 0}^{40248}\dfrac{1}{2k+1} \approx 6.283175 < 2\pi < 6.283188 \approx \displaystyle\sum_{k = 0}^{40249}\dfrac{1}{2k+1}$, the identity holds for $1 \le N \le 40248$ but fails for $N \ge 40249$.

The proof of this theorem has to do with Fourier transforms. Evaluating the Fourier transform of a function at $0$ gives you the integral of the function over $\mathbb{R}$. The Fourier transform of the product of sinc functions is a convolution of “rectangle” functions whose widths are proportional to $a_k$. If you convolve enough rectangle functions together, the value of the result at $0$ changes. This is an intuitive/non-rigorous explanation. I’m sure a more rigorous explanation can be found online.