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A major problem to be considered in this chapter is that of enumerating the isomorphism classes of objects in a given arithmetical category. It will be seen below that the Euler product formula for the zeta function provides a basic stepping-stone towards the solution of this problem, and that it leads to complete algebraic solutions for quite a number of specific categories of interest. § 1. A special algebra homomorphism The reader with some interest in algebra may have noticed quite early that much of our discussion of the Dirichlet algebra Dir (G) of an arithmetical semigroup G would be largely unaffected if complex-valued functions were replaced by functions f: G -k, where k is a general field with a real valuation I 1 (say).

I) For any aE G, d; (a) = d(a*) = 2"'(0). (ii) For any aE G, p(a) is equal to the total number of square-full divisors ~& 0 § 7. (-fonnulae In the preceding section, there was some discussion offunctionsfE Dir (G) such that f could be represented as a pseudo-convergent product of the form where the k, are complex numbers and the m, are positive integers. CH. 2. §7. 47 '-FORMULAE In the first place, examples appeared for which this product is finite and the k, are all rational integers. In such cases, we shall say that the function f possesses a finite '-formula.

Of a. (a) = d(a*) = 2"'(0). p/r, where (if one introduces the convention that 1*= I so that I is regarded as square-full) each p, = 0 or 2 -;§ p, -;§ OCi' This shows that the total number of square-full divisors of a is rt 1 OC 2 . ,. ocr = p (a ). 5. Proposition. (i) For any aE G, d; (a) = d(a*) = 2"'(0). (ii) For any aE G, p(a) is equal to the total number of square-full divisors ~& 0 § 7. (-fonnulae In the preceding section, there was some discussion offunctionsfE Dir (G) such that f could be represented as a pseudo-convergent product of the form where the k, are complex numbers and the m, are positive integers.

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