1.
Prove that \(e^{ix}=\cos(x)+i\sin(x)\) using Taylor series. Try to include proofs of the convergence of everything involved.
Exercises 25.9 Exercises
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.
10
activecalculus.org/single/sec-6-5-improper.html4.
Differentiate the function \(h(x)=x^x\text{.}\) Why is this question appropriate for this chapter?
5.
Verify numerically that \(\sum_{n=1}^\infty \frac{\mu(n)}{n}\to 0\text{;}\) first try a calculator, then a computer. How close can you get to zero before your computer gives up?
6.
Justify the comment in Remark 25.4.4. That is, if \(f,g\) have domain of the positive reals and both \(\sum_{n=1}^\infty f\left(x^{1/n}\right)/n\) and \(\sum_{n=1}^\infty g\left(x^{1/n}\right)/n\) converge absolutely, show that when \(g(x)=\sum_{n=1}^\infty f\left(x^{1/n}\right)/n\) we have \(f(x)=\sum_{n=1}^\infty \mu(n) g\left(x^{1/n}\right)/n\text{.}\) Hint: Substitute \(g\) into \(\sum_{m=1}^\infty \mu(m) g\left(x^{1/m}\right)/m\text{,}\) yielding a double sum in \(n,m\) with \(f\left(x^{1/mn}\right)\text{;}\) now carefully switch the sum to be over \(k=mn\) and \(d\mid k\text{,}\) ending with a sum where nearly everything cancels out due to Proposition 23.1.5.
7.
See Exercise Group 24.7.9–10 for the definition of \(\eta(s)\text{,}\) the Dirichlet eta function. Investigate whether there is a statement about this function which is logically equivalent to the Riemann hypothesis.
8.
Read one of the several excellent introductions to the Riemann Hypothesis intended for the “general reader”. (Some are listed in the Specialized References.)
Exercise Group.
A natural next direction to explore is the notion of elliptic curves. These exercises will help you think about what you find interesting about them!
9.
How are elliptic curves used in cryptography? (Peruse Chapters 11–12 for references.)
10.
Find out more about Mordell’s Theorem and its connection to this chapter and/or to Fermat’s Last Theorem.
11.
What is the Birch-Swinnerton-Dyer Conjecture? Find out as much about it as you can. (See the Specialized References, for instance.)
12.
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?
13.
What else do you want to know about numbers? What are you inspired to discover?
