189-249B: Honours Complex Variables

Practice Assignment 6

Not to be handed in.

A McGill regulation prevents graded assignments from being assigned so late in the term. The following are a few recommended exercises that you might find instructive as a complement to the material covered in class.

1. For further insights into the $\Gamma$-function and its behaviour, exercises 1-3 and 5 on pages 174-175 of Stein's book are recommended.

2. In our discussion of both the $\Gamma$ function and the Riemann zeta function, the Mellin transform plays a crucial role. Exercise 10 on page 177 and Exercise 17 on page 179 discuss it further are a good way to learn more about the Mellin transform.

3. Exercise 14 studies the rate of growth of $\log\Gamma(n)$ (and hence of $\log n!$) as $n$ goes to infinity.

4. In class, we essentially proved exercise 15 of page 178 expressing $\zeta(s)\Gamma(s)$ as the Mellin transorm of the function $\frac{1}{e^x-1}$.

It is worth working through this again if you missed it in class, and then ponder exercise 16 which explains how this observation can be parlayed into a proof of the meromorphic continuation of $\zeta(s)$ (with only a pole at $s=1$).

Food for thought: This approach to analytic continuation of $\zeta(s)$ is a bit simpler and more natural than the one expressing $\pi^{s/2} \Gamma(s/2) \zeta(s)$ as the Mellin transform of Riemann's theta function that we explained in class. What is the advantage of Riemann's approach?

Finally, if you felt inspired by the material, the four problems that start on page 179 might keep you pleasantly occupied, even after the course is over. I am particularly fond of exercise 4 giving a formula for $\zeta(s)$ at the even positive integers as an explicit rational multiple of the corresponding power of $\pi$, a substantial generalisation of Euler's identity $$ 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{25} + \cdots = \frac{\pi^2}{6},$$ which Euler derived himself in the same go, as a consequence of his factorisation of the sine function.

In a similar vein, use the functional equation to show that the value of $\zeta(s)$ at the negative odd integers is a rational number. In particular, derive Euler's famous (if slightly toungue in cheek) identity $$ 1+ 2 + 3 + 4 + 5 + \cdots = -\frac{1}{12}.$$