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The Product of All Primes is 4π^{2}

1+2+3+4+5+... = -1/12

and

1.2.3.4.5... = sqrt(2π)

Before you think I've lost my sanity, these results are examples of zeta regularisations which are frequently used in theoretical physics to 'renormalise' the infinite sums that have a habit of appearing there. Although their use in physics isn't terribly well justified, the zeta regularisation, due to Hawking, is well defined as a mathematical operation.

Methods for finding 'sums' of divergent series have a long pedigree. In fact, Hardy wrote an entire book on the subject. The techniques he uses range from Cesaro means to Borel summation.

Zeta regularised sums arise in the selection of the 'critical' dimensions in which various string theories work. For example the fact that bosonic string theory works in 26 dimensions can be seen as coming from 1+2+3+...=-1/12. Coming back down to Earth - a similar computation arises when computing the Casimir force, something that's actually measurable in a lab.

Labels: mathematics

## 3 Comments:

So, what is the sum of all primes? (Unfortunately the link to the pdf doesn't work for me.)

I don't know what the sum of all primes is, but I can show that the product of all primes has all the properties of zero, and the next numbers up and down are both prime (making the higher one higher than the highest known prime).

It can converge, but it doesn't always do that - it depends on whether e^n converges(that is how they changed the product to a summation in the formula). In case of primes it doesn't in the limit. because the product of the first n primes converges to e^n for n going to infinity.

It's not monotonic that is what it's saying - sometimes it has negative slope and sometimes positive and you do not know where it will go, but in case of primes it doesn't do that.

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