On 9/4/2019 8:39 AM, Bruno Marchal wrote:
On 2 Sep 2019, at 22:31, 'Brent Meeker' via Everything List
<[email protected]> wrote:
On 9/2/2019 8:57 AM, Bruno Marchal wrote:
Indeed that can be said illustrated from the fact that in string theory, to get
the mass of the photon right, we use 1+2+3+4+5+… = -1/12, which is a statement
about the prime numbers, just a little bit disguised
I'm sure you know that equation depends on assumptions about how limits are to
be taken.
Of course. That is the most axing part: that we need to invoque the prime
numbers, or equivalently the zeta function, to get a reasonable account of why
it make sense to say that the sum on all natural numbers is -1/12.
Now this “zeta-regularisation” has become common. To justify the mass of the
photon, the string theorits needs sophisticated tools (yet still essentially
arithmetical) from Number theory.
But it takes a redefinition of what one means by "convergence". There's
a good explanation here:
https://www.quora.com/Whats-the-intuition-behind-the-equation-1+2+3+-cdots-tfrac-1-12
Brent
It is essentially arithmetical by the fact that despite its use of complex
analysis in its most common formulation, the Riemann Hypothesis (Conjecture RH)
is an arithmetical sentence, indeed a pi_1 (the negation of a sigma_1
sentences).
If the RH is false, then even the tiny Robinson Arithmetic, which cannot even prove
that for all x 0 + x = x, is able to refute RH. That has been shown by Turing: if a
non trivial zero of zeta is not on the critical line, a machine can find it. The
existence of such zero is of (Post-Kleene)-complexity equal to “sigma_1".
Bruno
Brent
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