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J=1 Choose a generator gj for each of the factors Cnj and define characters on G by 2πi χ(j) (∗, . . , ∗, gj , ∗, . . , ∗) := e nj , i. e. ignore all entries except the j-th, and there use the same definition as in the Example above. Then the characters χ(1) , . . , χ(k) generate a subgroup ˆ which is isomorphic to G: Each χ(j) generates a cyclic group of order of G nj , and this group has trivial intersection with the span of all the other χ(i) ’s, since all characters in the latter have value 1 at gj .

Then the characters of G form a group under multiplication, (χ · ψ)(g) := χ(g)ψ(g), ˆ The identity in G ˆ is the trivial character. The group G ˆ is denoted G. isomorphic to G. In particular, any finite abelian group G of order n has exactly n distinct characters. 9 This is a lens, and through it you see into the ‘world of duality’. Think of this one day: What would happen if you took G = Z - what is ˆ := {homomorphisms Z → unit circle}? Z What if you took G = unit circle? The answer is: Fourier Analysis!

Riemann zeta function (updated by Roger Heath-Brown). Apostol is a little bit ponderous on this - he does the most general zeta function that he can. An easy example: Consider the function g(s) = 1 + s + s2 + . . The series converges for |s| < 1. Claim: g can be continued to a function which is analytic on the whole of C except for a simple pole at s = 1. Proof: −1 For |s| < 1, g(s) = s−1 . The latter expression is defined on C apart from a simple pole at s = 1 with residue −1. BUT: g is not defined by the series for |s| ≥ 1.