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It therefore seems interesting to ask whether any of the additional powers d k possess finite (-formulae. One might ask similar questions about other natural arithmetical functions. One method of approaching such problems stems from the next proposition and its corollary below. In order to state the proposition, first consider the subring Z [[t]] of C[[tl] consisting of all power series with rational integer coefficients. Then consider the nth cyclotomic polynomial C/J n = II (t-w), where the product ranges over all cp(n) primitive nth roots of unity ca.

Nfmn is pseudo-convergent. The assertion about repeated sums is easy to verify. 0 Z Although pseudo-convergence may be defined in terms of natural limits relative to the metric (}, the algebraic definition is especially useful later on. Similarly, although it is briefer to define pseudo-convergent double products in terms of limits as below, for the applications made later on it is sufficient to study products of the type considered in the next proposition. CH. 2. §3. ARITHMETICAL FUNCTIONS 32 n Firstly, letfmnE Dir (G), where m, n= 1,2, ....

Example: Pseudo-metrizable finite topological spaces. As an illustration of a situation in which the abstract "direct product" does not arise from the cartesian product of sets and the norm function arises differently from the previous cases, consider the category ~ of all pseudo-metrizable topological spaces offinite cardinal; here the Tj-axiom is not assumed. Clearly, a space X lies in ~ if and only if each connected component C of X lies in ~, and since there are only a finite number of such components C these must be both open and closed.

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