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In this way, one would obtain a generalized Dirichlet algebra Dir (G, k), whose properties might perhaps be worth studying in detail. In this book, it will not be necessary to go further into the above possibility, but instead it will be useful to consider the possibility of assuming slightly less about the initial semigroup G.

By the finiteness of the numbers NG(x) for x>O, it follows that M x = max {Nlal : lal ::§ x} -< oo, Therefore Fn(a)=F(a) for all aEG with lal::§x, whenever n~Mx' Thus (Fn-F»x for all n;§;Mx' and so Fn-F as n-=. This shows that the 0 metric space formed by Dir (G) relative to (} is complete. 1 is straightforward, and wiII be left as an exercise. 0 In order to define infinite products in Dir (G), the simplest procedure appears to be to call a product JI:=lf" (f"EDir (G») pseudo-convergent if and only if JI:= 1 f" tends to some limit F as N - =, relative to the metric (!

Suppose that fE Dir (G) can be represented as a pseudoconvergent product of the form where the k, are complex numbers and the m, are positive integers. Then, for any completely multiplicative function XEDir (G), fez, X) = II [(G (m,z, xm')]k'. 4. Corollary. Let X denote a completely multiplicative function on G. Then: (i) For any positive integer m, dk(mz, Xm) = [(G(mz, Xm)]k. ARITHMETICAL FUNCTIONS 46 (ii) (iii) (iv) (v) (vi) (vii) CH. 2. §7. d 2(z, X) = [(G (z, X)]4J(G (2z, l). d. (z, X) = [(G(z, X)]2J(G (2z, X2), P(z, X) = (a

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Abstract analytic number theory. V12 by Knopfmacher

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