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Extra info for Abstract analytic number theory. V12
At the appropriate later stage in this chapter, the reader may find it interesting to compare the present methods with those given in Chapter XVII of Hardy and Wright [I] for the semigroup Gz of positive integers; the latter methods make use of ordinary analysis, but Hardy and Wright also make some remarks about the possible use of formal series in deriving certain identities. ) The following proposition assembles a few elementary facts about pseudoconvergent sums and products, sufficient for most purposes below.
2 implies that it satisfies the Euler product formula To a large extent, the importance and usefulness of (G will be seen to stem from this formula, which may be regarded as a formal analytic translation CH. 2. §5. THE ZETA AND M6BlUS FUNCTIONS 37 of the unique factorization property of G. In particular, the Euler product formula provides the first step in solving certain enumeration problems for an arithmetical category Lemma. (d) log Idl; A (d). Proof. 3. (d) if (a,b) = I, otherwise. (d) Ibl~:al dla,dlb Z Jl(d) Z 1 = Z Jl(d) L I dla Ibl lal dla Icdl lal ~ dlb ~ CH. 2. §6. FURTHER NATURAL ARITHMETICAL FUNCfIONS 45 This proves (i), For (iv), consider an element a=P11Z'P21Z2 . . p/", where the PIEP are distinct and lXe·O. Pli = p,la r IZ, Z Z log \Pil = log lal· i=1 s=1 o This lemma shows that
Lemma. (d) log Idl; A (d). Proof. 3. (d) if (a,b) = I, otherwise. (d) Ibl~:al dla,dlb Z Jl(d) Z 1 = Z Jl(d) L I dla Ibl lal dla Icdl lal ~ dlb ~ CH. 2. §6. FURTHER NATURAL ARITHMETICAL FUNCfIONS 45 This proves (i), For (iv), consider an element a=P11Z'P21Z2 . . p/", where the PIEP are distinct and lXe·O. Pli = p,la r IZ, Z Z log \Pil = log lal· i=1 s=1 o This lemma shows that