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Number Theory (MA3Z7)
Problem Sheet VI
1. Let Q ∈ R. Find the generating function of σα (n) =Σd|ndα .
2. Prove that
for s > max{cf , cg } with cf and cg the abscissa of absolute con- vergence of Σ f(n)n—s and Σ∞n=1 g(n)/ns respectively, where h is given by
the Dirichlet convolution of f and g (see Problems IV, Question 2).
3. Show that
has abscissa of convergence equal to 0 and abscissa of absolute convergence 1.
Further, prove that for s > 1,
4. Let g(n) =Σdjn μ(d)2 . Show that
(i) g(n) is multiplicative;
(ii) g(pk ) = 2 for p prime and k ∈ N;
(iii) g(n) = 2ω(n), where ω(n) is the number of distinct prime factors of n;
(iv) the generating function of g(n) is 
5. Show that if f is completely multiplicative, then
for s > abscissa of absolute convergence of the LHS series.