## Regular aggregation sequences and their (in)finite series

REF. DATE:11 October 2015
CREATED/MODIFIED: 13 October 2015
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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…

E. Rocha (author)said: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

(1)

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

(2)

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

where .

(5) We have . Moreover, if is a decomposition into prime numbers, then

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