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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 (! , the product is convergent relative to (}). Then F may be called the value of the infinite product. For some purposes, it might be convenient to exclude the value 0, as in ordinary analysis, but for present considerations this is immaterial because the later discussion wiII be concerned exclusively with products of the following type.

In such a case, we shall simply say thatfpossesses a '-formula. Finally, iff admits a general representation of the above form, we shall say that f possesses a generalized '-formula. The significance of the last concept is indicated by the next theorem. 1. Theorem. A function fEDir (G) is a PIM-function if and only if f possesses a generalized '-formula. IIj Proof. In § 6 and in the above definitions, the products were written as though the indices i were ranging over a single sequence of values.

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. It is well known (see for example Lang [1]) that the cyclotomic polynomials have rational integer coefficients, are irreducible over the field Q of rational numbers, and satisfy the inversion formulae C/Jn = II (td_lyt(n/d), din where Jl now denotes the Mobius function on the semigroup Gz . 3 shows that the constant term of C/Jn is II din (_I)Il(n/d) = II (_lll(d) = (_I)Edlnll(d) din if n = I, otherwise.

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Abstract analytic number theory. V12 by Knopfmacher

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