Is there a source, of proof of this? I have no idea how to do those integrals.
Apparently there's a new, pretty cool physical interpretation of the Borwein integrals in terms of random walks:
In the new paper, the physicists show that the movements of infinitely many random walkers can be used to model the emergence and disappearance of the patterns in the Borwein integrals. To begin, the random walkers all start at the point zero on the one-dimensional number line. For the first step, each walker is allowed to move a random distance of up to 1 unit, either left or right. For the second step, each walker may move a random distance of up to 1/3, then a random distance of up to 1/5, then 1/7, 1/9, etc. That is, each successive allowable step distance corresponds to the next value of the expression 1/(2n—1).
The main question is, what is the fraction of random walkers at the starting point (the origin) after each time step? It turns out that the fraction (more precisely, the probability density) of walkers at the origin at each time step n corresponds to the solution to the Borwein integral using the same n value.
As the physicists explain, for the first seven steps, the probability density that a walker ends up at the origin is always ½, which via the theorem above corresponds to an integral value of π. The key idea is that, up to this time, the density of walkers at the origin is the same as if the entire number line was uniformly populated with walkers. In reality, as the maximum distance of each step is restricted, only part of the number line is accessible, i.e., the walkers' world is finite.
However, for the first seven steps, the walkers at the origin perceive that their world is infinite, since they do not possess any information about the existence of boundaries that would indicate that the world is finite. This is because none of those walkers that reached the outer boundary of their world (+1 or -1 after the first step) would have been able to make it back to the starting point in less than seven steps, even if taking the maximum size steps allowed and all in the direction toward the starting point. As these walkers had zero probability of showing up at the starting point before the eighth step, they could not affect the fraction of random walkers at the starting point. So for the first seven steps, the density of walkers at the origin is fixed at ½ (it is "protected").
But once those walkers that have reached +1 or -1 return to the origin, the situation changes. After the eighth step, it's possible that some of these walkers return to the starting point. Now these walkers act as "messengers" in the sense that their return to the starting point reveals the existence of a boundary, telling the other walkers at the origin that their world is finite, and therefore influencing the density of walkers at the origin.
So basically, if I understand it correctly, in the first step, you send out a "messenger" from the origin along the real line, which ends up at a random x coordinate between -1 and 1 (e.g. at 0.437737). Assume it ends up at x = 1 after one step. In the next step, this messenger moves up to 1/3 units (randomly) either left or right, e.g. between x = 1-1/3 = 2/3 and x = 1+1/3 = 4/3. Assume it ends up at x = 2/3. In the third step, it moves up to 1/5 units either left or right. Assume it ends up at x = 2/3 - 1/5 = 7/15 ≈ 0.46666...
Continuing in this fashion, and since
1 > 1/3
1 > 1/3 + 1/5
1 > 1/3 + 1/5 + 1/7
1 > 1/3 + 1/5 + 1/7 + 1/9
1 > 1/3 + 1/5 + 1/7 + 1/9 + 1/11
1 > 1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/13
1 < 1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/13 + 1/15,
what happens is effectively that a messenger that was initially sent out to x = 1 (or x = -1) is able to return to the origin after 8 steps to inform it (or rather, another messenger at the origin) that the "probability
" of ending up at the origin is skewed away from what it "thinks" it is, since the walkers (unbeknownst to it) have restrictions on their step lengths.
Similar arguments based on causality or limits on information propagation speed can be used to evaluate a number of tricky integrals; see the arXiv paper
As a side note, the above reminds me of a recent, small modification of the well-studied Pólya urn model
that seems to describe how patterns in innovation (and Zipf's, Heaps' and Taylor's laws) arise: https://www.technologyreview.com/s/6...vations-arise/
It's pretty interesting how small (and quite intuitive, in retrospect, IMAO) changes to simple models can give rise to explanations that have eluded mathematicians for many years. It makes me think there's still a lot of room for messing around with formulas, axioms and ideas and looking at the resulting patterns numerically in order to make very significant discoveries.