I like the explanation here distinguishing between the Reimann and Euler zeta functions, but I am still skeptical about their use in characterizing divergent sums.
Even though the Reimann zeta function gives a value for the attraction between two metal plates in a vacuum that matches the value given my QM, it still does not make the sum of the natural numbers or QM any more real. Sure, the both have utility in explaining things: i.e. the force between two plates, but that does not definitively exclude other possibilities.

Galileo sort of argued this for planetary motion to appease the church: you don't have to really believe that the Earth revolves around the sun, but if you want predict where planets will be, you can do the math as if the Earth does revolve around the sun.
And Einstein's general relativity: An elevator sitting in a gravitational field behaves as if we are in an elevator in an accelerating spaceship. Since there is no distinguishing these two scenarios, they are equivalent.
Finally, if Fourier series can represent other functions (I am thinking about X-ray crystallography) then which is more real: the Fourier co-efficients or the values of the real function. If they can both represent the same thing, but it is useful to use one sometimes and the other at other times, can one be more real than the other?
Back to QM and the sum or natural numbers... I wonder whether the treatment of the problem is incorrect (unreal) in both cases, but it cancels each other out to arrive at accurate values. Aren't there other theories still around that describe subatomic events without invoking QM? If these are applied to the Casimir plates, maybe it bypasses the Euler zeta function and uses the Reimann zeta function.

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