Mathematics 5
Analytic Number Theory
Spring 2025
Assignment 3
Please hand in by 12 noon on Friday, 14 March
1. Let χ be a Dirichlet character mod q and consider its theta function
Note that χ(−1)2 = χ(1) = 1 and so χ(−1) ∈ {−1, 1}. We say χ is even if χ(−1) = 1. We say χ is odd if χ(−1) = −1.
(a) Show that if χ is odd, then ϑ(t; χ) ≡ 0 and if χ is even, then
(b) If χ is any Dirichlet character mod q, show that
Hint: You may find the elementary inequaltiy ex − 1 ≥ x for all x > 0 useful.
2. Let χ be a an even Dirichlet character mod q. Recall the L function defined by χ is given by
Show that for every n ≥ 1,
Sum over n ≥ 1 to conclude that for Re z > 1,
Hint: To justify interchanging the sum and integral, you can use the following analysis result : if where then
For the remaining part of the assignment, we will assume that χ is an even Dirichlet character mod q whose theta function
satisfies the following functional equation:
where
3. Split the integral in (†) so that for Re z > 1,
Use the functional equation (1) so show that for Re z > 1,
Show that the sum of the integrals on the right defines an analytic function on the whole complex plane C. This gives the analytic continuation of L(z; χ).
4. Suppose that χ is an even Dirichlet character mod q where q is a prime. In this case, the sum cχ,1 in the red box satisfies (you may assume this).
Apply the previous question to the even character χ to show that
This is the functional equation for L(z; χ) when χ is an even character.