Number Theory: In Context and Interactive

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 $\zeta (-1)=\frac{1}{12}\text{.}$ 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. You may wish to have a refresher from any calculus textbook.

4.

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

5.

Verify numerically that $\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n}\to 0\text{;}$ first try a calculator, then a 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.)

10.

Answer one of these questions, or all of them.

• What is a partition of a number?

• What are continued fractions?

• What is a number field?

11.

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