Motivated by the (renormalization) of some classical divergent series in String Theory, e.g.
(see here), I start thinking on the convergence meaning of a class of divergent series, trying to make sense of them without following some of the (standard) approaches as Hardy resummation or Zeta function regularization (e.g. wikipedia). The main idea was to see if it is possible to define a class of sequences and a equivalence operation, by aggregating terms, such that their “value” is determined by some constant sequence in the same equivalent class.
In the comments of this post, you see the precise notions and a preliminary result, which is quite open for discussion…
NOTATION. I use to denote a tupple (e.g. a 2-tupple), of real numbers, since means a set (i.e. no order and no element repetition) and is an open interval. We will also consider tupples of functions. The size of a tupple is denoted by . Any sequence (an infinite-tupple) can be extended by zeros on the left, so it is infinite in the two directions. Therefore the index of a sequence is an element of , where the first element of a sequence is denoted by . Moreover, since for all , the relevant set of indices is . For , we denote by the integers that are congruent to , i.e. the set .
Let , , and . Define the conditional function as
which, for convenience, we just write as
DEFINITION. A regular aggregation sequence (r.a.s.) is a sequence such that exist fixed tupples and , with , and , such that
We call the pair a realization of u, which is not unique for each sequence u. The set of all r.a.s. is denoted by .
DEFINITION. The aggregation operation , with and , is the map such that,
LEMMA. For and , the map has the following properties:
(1) It is well-defined, i.e. .
(2) We have
where , , and
is a bijection.
(3) Fix and define by
The function is an homomorphism.
(4) Let , and , then
(5) We have . Moreover, if is a decomposition into prime numbers, then
See the full content in this Mathdir topic.