Abstract analytic number theory, Volume 12: V12 by Knopfmacher

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Additional info for Abstract analytic number theory, Volume 12: V12 (North-Holland Mathematical Library)

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0 otherwise. The result therefore follows. 0 Now let X denote a completely multiplicative function on G. L: C1EG wherefx(a)=x(a)f(a) for aEG. fx) defines a continuous algebra endomorphism of the Dirichlet algebra Dir (G). Further, if X(a) ~O for all aEG, then this endomorphism is a topological algebra automorphism of Dir (G). In the general case, the rule f-fx also induces group endomorphisms of the group of units of Dir (G) and of M (G); again, these are automorphisms if x(a)~O for all aEG.

Then Il:=l/"(z)gn(z) is a pseudo-convergent product and (iv) Let Il:=l/" be a pseudo-convergent product in Dir (G) such that /,,(1)= I for each n= 1,2, .... Then Il:=l [/,,(Z)]-l is a pseudo-convergent product and The proof of this proposition is straightforward, especially if one first notes the next lemma, which may be proved by elementary arguments of a standard type; the details are left as an exercise. 5. Lemma. (i) Let ft,f2' ... and gt> g2'''' be convergent sequences in Dir (G), and let f = lim n_ ec / " , g = limn_ ~ gn' Then the following limits exist in Dir (G) and have the values indicated: lim fn(z)gn(z) = f(z)g(z).

Then Il:=l/"(z)gn(z) is a pseudo-convergent product and (iv) Let Il:=l/" be a pseudo-convergent product in Dir (G) such that /,,(1)= I for each n= 1,2, .... Then Il:=l [/,,(Z)]-l is a pseudo-convergent product and The proof of this proposition is straightforward, especially if one first notes the next lemma, which may be proved by elementary arguments of a standard type; the details are left as an exercise. 5. Lemma. (i) Let ft,f2' ... and gt> g2'''' be convergent sequences in Dir (G), and let f = lim n_ ec / " , g = limn_ ~ gn' Then the following limits exist in Dir (G) and have the values indicated: lim fn(z)gn(z) = f(z)g(z).

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