The NUMDAM Project is digitizing the archives of French mathematical journals, and placing them online. Some of the journals are quite venerable — the Annales Scientifiques de l’École Normale Supérieure was founded in 1864. Another highlight is the prestigious Publications Mathématiques de l’IHÉS.

I was looking at the Wikipedia entry for Harmonic number, where I spotted a rather surprising reformulation of the Riemann hypothesis.

The Riemann hypothesis was already known to be equivalent to a not-very-complicated statement about the distribution of primes. Let π be the number of primes less than n. Then the Riemann hypothesis is equivalent to:

for all ε > 0. This fact, which goes back at least to Riemann, is the main reason why the Riemann hypothesis is of interest. In 2002, Jeffrey Lagarias found an even more elementary statement.:

where H_n is the nth Harmonic number (the sum of reciprocals less than or equal to n). It almost looks you could solve it, doesn’t it?

IBM Research has put up the July challenge on their Ponder This site.

Update: Here is the problem:

Upon a rectangular table of finite dimensions L by W, we place n identical, circular coins; some of the coins may be not entirely on the table, and some may overlap. The placement is such that no new coin can be added (with its center on the table) without overlapping one of the old coins. Prove that the entire surface of the table can be covered completely by 4n coins.

Since most posts don’t get many comments, I thought I would make one the required audience participation. The subject is “fundamental” theorems in the various subjects. What I am going for is hard to actually describe, but encapsulates a theorem being fundamental, its utility, its depth. It is the result in the subject that would hurt the most not to have, but does not have to be the putative “fundamental theorem of X”

For example, my votes for a few subjects:

Calculus: Mean Value Theorem.
Probability: Linearity of expected value.
Model Theory: The compactness theorem.

Allen Hatcher of Cornell appears to be undertaking the quixotic goal of writing accessible “introductory” textbooks for the entirety of Algebraic Topology. The first volume is already the definitive introductory work in the subject, covering the Fundamental Group, Homology, Cohomology, and Homotopy and is available online.

The style is geometric, so those whose nascent views on Algebraic Topology are functorial may be better served by Rotman’s book, but it is hard to recommend highly enough since Hatcher is an excellent expositor, and the book is clearly written for students rather than a vanity piece aimed at colleagues, a major pet peeve of mine.

Urs Lang has posted some lecture notes on geometric measure theory.

I’ve been reading Fremlin’s book, and I’ve seen a very surprising theorem that was new to me: Maharam’s Theorem. If you take an set of coins, you can define a measure space on the set of coin flips by taking the product measure. This is a probability measure: the measure of a set is the probability of a coin flip appearing in that set. Since it is a probability measure, it’s well-defined for sets of every cardinality.

You can combine any two measure spaces by taking their disjoint union; the measures are combined by addition. More generally, you can take a weighted sum. Maharam’s theorem states that every nontrivial complete measure space can be constructed from sets of coin flips by taking weighted sums. For example, counting measure is an infinite sum of flips of a single coin. Lebesgue measure on the unit interval arises from flipping an infinite number of coins.

This means that there are not very many types of (complete) measure spaces.

David Fremlin is writing the definitive tome on measure theory. He completed four volumes and is currently working on a fifth. You can find it all at the book’s web site.

For a more succinct account of measure theory, see James Hirschorn’s survey article.