It's easy to write nonsense, not so easy to make sense.
1+2+3+4+... is a divergent series-the result is +infinity.
There are many circumstances, however, when it can be proved that what is meant by that sum is not +infinity, but, rather, ζ(-1), where ζ(s) is the Riemann zeta function. This means that the appearance of the sum itself is an indication of an error: The result of the calculation is, in fact, finite and the non-trivial statement is that it is equal to ζ(-1). Unfortunately many accounts simply stop at the ``regularization'' and don't explain how it comes about.
This paper: https://arxiv.org/pdf/1607.06493.pdf might be a good place to start.
The Riemann zeta function, when Re (s) > 1, can, in turn, be proved to be given by the expression 1+2-s+3-s+4-s+... This series, additionally, does NOT converge when s=-1. On the other hand, it can be proved that ζ(-1)=-1/12. This has led to many ``jokes'' that 1+2+3+4+...=-1/12, which is, of course, wrong. The ``jokes'' hide the interesting part, that the calculation that led to the appearance of the divergent series is wrong and that the correct calculation involves replacing the divergent series by ζ(-1). That ζ(-1) might be thought as the sum 1+2+3+4+... is, simply, wrong.
Similar ``jokes'' can be made with the sums of infinite subsets of integers.
The function, whose representation when Re(s)>1 is given by 1+3-s+5-s+7-s+... i.e. over the primes, is known as the ``prime zeta function''. One relation with the Riemann zeta function is explored here: Preprint Prime zeta function statistics and Riemann zero-difference repulsion
Instead of repeating the ``jokes'' of people who should know better, it would be more useful to actually understand when ``zeta function regularization''
is appropriate.
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