Possible essay topic: Generalized Bernoulli numbers
If you're interested in writing up an essay for the PeakMath Essay Competition and you're looking for something cool to write about, one suggestion is to learn about Bernoulli numbers, Bernoulli polynomials, and generalized Bernoulli numbers, and write up an exposition of some research paper based on these concepts. The generalized Bernoulli numbers are related to special values of Dirichlet L-functions, and studying them is a pathway to a host of active and interesting research directions, such as $p$-adic L-functions and class numbers of quadratic number fields. Famously, Ada Lovelace's "first ever computer algorithm" was written to compute Bernoulli numbers, and you can find some fun details on this at 101 Computing.
Some introductory references, to get started. All of these are great!
- The Bernoulli number page
- An introductory note by Jake Huryn
- A brief primer, by Nathaniel Larson
- Various blog posts of John Baez (start here and follow the links)
- Barry Mazur: Bernoulli numbers and the unity of mathematics. These 19 pages of (rough but very beautiful) notes seem to have disappeared from the internet, but I found a copy at the Wayback Machine. They are much more interesting than the short 4-page handout with the same title.
- A primer by Tom Apostol (paywalled, but you can read it with a Jstor account, and a copy also seems to be available on Scribd)
- The Bernoulli manifesto
Some possible topics and papers you could pursue for an essay:
- Henri Cohen has a wonderful paper on different methods for computing special values of Dirichlet L-functions. Any of these methods would be an interesting topic to investigate, for example the strange connection to Eisenstein series.
- The Riemann Hypothesis can be formulated in terms of Bernoulli numbers via the Baéz-Duarte criterion. A very nice exposition of this is found in Maslanka, and you can also find other more recent articles on the subject. One thing I don't know is if this formulation of RH can be generalized to Dirichlet L-functions or perhaps even L-functions in general.
- Another approach to RH via Bernoulli numbers can be found in a series of papers by Voros, in which he (roughly speaking) constructs a "discretized" version of the Keiper-Li coefficients, whose positivity is equivalent to RH. Check out these papers on arxiv, for example here and here.
- Don Zagier: Curious and exotic identities for Bernoulli numbers. This is an appendix in a wonderful book by Arakawa, Ibukiyama and Kaneko, where you will also find an exposition of many other topics, like class numbers, connections to Stirling numbers, Dirichlet L-functions, and exponential sums.
I'll end with an excerpt from Richard Feynman's Lectures on Computation which captures perfectly the spirit of the essay competition, in describing the practice of figuring things out on your own that may already have been found by others before you, until you get to the point where you transcend the boundary to the unknown.
