Mathematics 5 Analytic Number Theory Spring 2025 3

Java Python 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,

Hdai 写Mathematics 5 Analytic Number Theory Spring 2025 Assignment 3 int: 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