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Homework 2. Due by February, 21.

1. Dirichlet theorem implies the following statement: if h and k are any two integers with (h, k) = 1, then there exists at least one prime number of the formkn+h. Prove that this statement also implies Dirichlet’s theorem.

2. Prove that the function f(t) =e−πt2 is its own Fourier transform.

3. Let θ(t) =P

n∈Ze−πn2t. For t >0 prove that θ(t) = 1

√tθ(1 t).

4. Compute: a) Γ(12); b)ξ(1); c)ξ(0).

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