By Graham Everest

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13 If you remember something about Fourier Analysis - this is really like Fourier Analysis, only nicer. The rule 1 f(g)g(h) |G| h∈G is an inner product on the vector space of all functions on G, and the characters form a complete orthonormal set. There are no worries about convergence, integrability . . And in particular, any complex function on G can be written as linear combination of the characters. 14 ˆ Extend χ Given 1 < q ∈ N, consider G := U (Z/q) and a character χ ∈ G. to a function X on N by setting X(n) := χ(n mod q) if n coprime to q 0 otherwise (105) Then the function X is called a Dirichlet character modulo q.

Use reasoning by absurdity and N := 2 · 3 · . . · pr + 5. What do you take for primes p ≡ 1 (mod 6) (only for 1 and 5 modulo 6, there is a chance of infinitely many primes)? Exercise. Isn’t this nice?? NO! The results are nice, but the proofs are awkward. We would like to have a general principle for proving such results. 2 along the lines of proof 2 for the infinity of primes. Consider for odd n ∈ N the function 1 + (−1) c1 (n) := 2 n−1 2 1 for n ≡ 1 0 for n ≡ 3 = (mod 4) (mod 4) (87) This is a gadget for picking out a particular congruence class.

14 ˆ Extend χ Given 1 < q ∈ N, consider G := U (Z/q) and a character χ ∈ G. to a function X on N by setting X(n) := χ(n mod q) if n coprime to q 0 otherwise (105) Then the function X is called a Dirichlet character modulo q. Note that this is a slight abuse of language - it is not meant to say that N were a group. However, we will even write χ instead of X for the Dirichlet character associated to χ.

### Analytic Number Theory by Graham Everest

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