I am reading “Prime Obsession” by John Derbyshire.
It is about the Riemann hypothesis and includes a great deal of the history surrounding its origin.
The provided historical background gives a feeling of what it was like to be a mathematician in 18’th and 19’th century Europe.
It assumes some algebra and recommends a dash of calculus.
Derbyshire conveys a mathematician’s sense of what a beautiful question or result is.
The book takes the reader thru some non trivial math, such as convergence of series, most gracefully.
I am most pleased to hear of Nicol d’Oresme’s 12th century proof that the harmonic series diverges.
It is an elegant proof which is better than most of those discovered later.
If you subdivide the series into consecutive groups whose sizes are 1, 2, 4, 8 etc, then the sum of each subgroup is at least 1/2.
The whole series thus diverges.
Critical to the story is that Σn−2 converges.
Derbyshire does not give a proof of this but d’Oresme’s proof plan can be easily extended thus:
With the same grouping of terms, the g’th group can be seen to sum to something less than 2−g.
The whole series thus converges.
Where the book concerns non-trivial material, it leads you thru some of the surface math that is accessible to the clever high-school graduate.
Many of the omissions are filled in by an undergraduate math courses.
The outlines of the proof are in view much more so than any math popularization that I can recall.
I stumbled across a paper by Odlyzko a few years ago on the spectrum of random orthogonal matrices.
These eigenvalues are on the unit circle and also display the repulsion effect.
I know what a random orthogonal matrix means:
It denotes a randomly selected orientation (or rotation) of rigid n dimensional ball with fixed center.
This is a probability distribution addressed by Haar measures on compact Lie groups.
Each eigenvalue is associated with a ‘pure rotation’ about some n−2 dimensional subspace which is left fixed by the pure rotation.
Parts of the ball in that subspace are fixed by the pure rotation.
The full rotation may be composed of these pure rotations.
The amount of the pure rotation is the argument of the eigenvalue.
When two eigenvalues are equal (degenerate case) we have a fixed n−4 dimensional space and there are many pure rotation pairs that can be paired to give the effect of this 4D rotation.
They commute.
In just 4D this is the famous rotation that produces parallel orbits that do not intersect and ‘fiber the surface’.
Degenerate orthogonal matrices are of measure 0.
I suspect that nearly degenerate matrices (near eigenvalues) are likewise scarce, thus the repulsion effect.
Between pages 297 and 328, pages numbered 297+4n and 300+4n are blank and lack even page numbers.
Fragmentary English sentences appear at the edges of these lacunae.
I noticed the hole when I looked up the portion on p-adic numbers.
The paperback version does not suffer this problem.