1

Prove that \(e^{ix}=\cos(x)+i\sin(x)\) using Taylor series. Try to include proofs of the convergence of everything involved.

2

Many books have a chain of reasoning interpreting the value \(\zeta(-1)=\frac{1}{12}\). Find a physical one and summarize the argument. (The Specialized References and Other References may have some suggestions.) Do you buy that adding all positive integers could possibly have a meaning?

3

Show all details for the improper integrals in Section 25.5.

4

Differentiate the function \(h(x)=x^x\). Why is this question appropriate for this chapter?

5

Verify numerically that \(\sum_{n=1}^\infty \frac{\mu(n)}{n}\to 0\) – by calculator, then by computer. How close can you get to zero before your computer gives up?

6

Read one of the several excellent introductions to the Riemann Hypothesis intended for the “general reader”. (Some are listed in the Specialized References.)

7

What is the Birch-Swinnerton-Dyer Conjecture? Find out as much about it as you can.

8

Answer one of these questions, or all of them.

• What are partitions of a number?

• What are continued fractions?

• What is an elliptic curve, and how is it used in cryptography?

• What is a number field?

9

What else do you want to know about numbers? What are you inspired to discover?