# References for mathematical theory of summability of divergent series

Once in a while, I can't help it to ask very broad questions. I have read (a portion of) Hardy's Divergent Series. Back then, I think besides in mathematics, divergent series and the need to assign values to them hadn't arisen. But nowadays, for example, almost all explanations of the 26 dimensions of Bosonic string theory deals with such an assignment.

Also, the internet literally exploded on Youtube explaining these kinds of issues (they conveniently left out all the proofs). Questions:

• Are there (more modern) references for the mathematical theory of the summability of divergent series, and its philosophy (I.e. usage of such things in physics) ?
• I have seen divergent series of real numbers sum to a complex number with non-zero imaginary part. Is there something more to it (for example, are there known divergent series that lead to a quaternion) ?

I am tagging this with meta, biglist and analysis, for obvious reasons.