Is there a reason that a separate symbol and term are not used for this different form of summation?

Mainly that in physics, these sums crop up naturally, using normal summations over real numbers, often sums of energies over different modes.

For instance, if you look at an idealized quantum violin string, each mode of vibration has a minimum energy (zero point energy) proportional to the frequency of vibration - classically they can all be 0, but not quantum mechanically. When you try to ask, then, what the minimum energy of the string is, you end up with a term that is literally 1+2+3+... and no particular reason to intepret those as complex or anything. But in a lot of ways, if you just barrel through and treat it as if you can do the sum, you get real results - the Casimir effect is an example where a real force can be predicted and measured based on calculations that are zeta regularized.

It's also worth noting that the dimensionalities in various string theories tend to hinge on the exact values of these infinite sums. Bosonic string theory being 26 dimensional comes out of, IIRC, a consistency equation that ends up including -1/12 because of a 1+2+3... sum. If memory serves, that result can be rigorously established in other ways, as well.

Thanks for the insight. I'll have to look at this more closely sometime... not sure I can grok it casually.

I've been watching the PBS spacetime videos on youtube a lot lately. The recent ones have been diving into zero-point energy (and related stuff), which has been fascinating.

Some people say that they are using "=" to mean something else, and relying on context to distinguish which "=" they mean. But it's less fun to say that the sum of the natural numbers "is associated with" -1/12, especially if you have to say how, exactly.