Use hypotheses (ii) and (iii) of Definition 11, rather than hypothesis (i). (iii) gives the case when q is prime, and (ii) then gives the general squarefree case as a consequence.

More generally, bounds on multiplicative functions should usually be proven by first considering the prime case (or the prime power case, if one is not restricted to squarefree numbers).

]]>One can bound by and by .

]]>Hmm, that is a numbering issue; one can use for instance Lemma 8 from the previous post. (These estimates can also be found in standard multiplicative number theory texts, e.g. Corollary 2.15 of Montgomery-Vaughan. A good rule of thumb regarding these sorts of estimates is that the number of divisors of a large number is typically of size ; this is closely related to the Erdos-Kac theorem, which is a precise version of the assertion that the number of prime factors of a number is typically of size .)

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