
Breakdown: on Borwein integrals - benbreen
https://www.futilitycloset.com/2018/02/02/breakdown-2/
======
lake99
I saw this in one of the most famous answers on Mathoverflow [1]. The CAS
expert (Jacques Carette) who wasted three days trying to locate the "problem"
in the software chimes in in the comments.

[1] [https://mathoverflow.net/questions/11517/computer-algebra-
er...](https://mathoverflow.net/questions/11517/computer-algebra-errors)

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ocfnash
The poor Maple developer who tried to fix the "bug" is active on MathOverflow
and commented briefly on his experience of this in a comment below this
answer: [https://mathoverflow.net/a/11607](https://mathoverflow.net/a/11607)

------
twtw
Unfortunately I can't think through it now, but I would bet there is a fun
connection to be made with 3*5=15. Something about roots more frequently
getting repeated, or zeros in the s-plane going to a higher order maybe? I'd
be delighted if someone can chime in with the connection, if there is one.

~~~
SidiousL
You can compute this integral as follows. First, you remember that sinc(x) =
sin(x)/x is the Fourier transform of 1/2 * indicator function of [-1, 1]. In
general you have:

    
    
      sinc(x/n) = n/2 \int_{-1/n}^{1/n} d t e^{i t x}.
    

Next, notice that

    
    
      \int_0^\infty sinc(x) \dots sinc(x/n) = 1/2 \int_{-\infty}^\infty sinc(x) \dots sinc(x/n).
    
    

Next, replace the sinc functions by their Fourier representation, change the
order of integration and use the formula

    
    
      \int_{-\infty}^\infty d x e^{i a x} = 2 pi \delta(a),
    

where \delta(a) is the Dirac delta. You are left with an integral of a delta
function over a product of intervals. What remains to be done is just some
tedious computation.

\int_{-1}^1 d t_0 \dots \int_{-1/(2 k + 1)}^{1/(2 k + 1)} d t_k 1/2 \dots (2 k
+ 1)/2 2 \pi \delta(t_1 + \dots + t_k).

Next, the question is whether t_1 + \dots + t_k = 0 has a solution for t's in
the integration domain. This holds until k = 7. To see this, compute

1/3 + 1/5 = 8/15 < 1

1/3 + 1/5 + 1/7 = 71/105 < 1

1/3 + 1/5 + 1/7 + 1/9 = 248/315 < 1

1/3 + 1/5 + 1/7 + 1/9 + 1/11 = 3043/3465 <1

1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/13 = 43024/45045 < 1

but

1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/13 + 1/15 = 46027/45045 > 1.

