Author: Kowalski

This is what H. Widom understandably says of the formula

$latex \frac{G(1+\alpha)^2}{G(1+2\alpha)}=e^{\alpha(\alpha-1)/2}\pi^{\alpha/2}2^{-2\alpha^2+\alpha}\Gamma(\alpha+1/2)^{-\alpha}\prod_{j\geq 1}{\frac{\pi}{2}x_j(\alpha)J_{\alpha+1/2}(x_j(\alpha))^2}$

proved in his 1973 paper Toeplitz Determinants with Singular Generating Functions. Here the variable α is a complex number, G is the Barnes function, which can be expressed as

$latex G(z+1)=(2\pi)^{z/2}e^{-(z(z+1)+\gamma z^2)/2}\prod_{n\geq 1}{(1+z/n)^n\exp(-z+z^2/(2n)}}$

J_ν(z) denotes the standard Bessel functions of the first kind, and the x_j(α) are the zeros with positive real part of the Bessel function

$latex J_{\alpha-1/2}(z).$

Widom adds that this formula is “established here by what is certainly a roundabout procedure”. In fact, he obtains it by comparing two independent computations of, well, Toeplitz determinants with singular generating functions (one his own, one a special case due to Lenard), and it is hard to imagine finding an identity like that by chance… It seems to be completely and wonderfully different from most other analytic identities which are commonly known!

In one of the last sections of his paper, Widom sketches a direct proof, which is roughly as follows: one shows that the ratio of the two quantities is an entire function of α of order (at most) 2; it follows that it must be of the form

$latex e^{a+bz+cz^2}$

for some constants a, b and c. Hence checking the formula for three values of α will prove it for all. For α=0, this is not difficult; for α=2, it boils down to the elementary formula

$latex \prod_{j\geq 1}{\sin^2 x_j}=\frac{e}{3}$

where now x_j are the positive zeros of the function

$latex f(x)=\frac{1}{x}\sin x-\cos x.$

Widom gives a very nice proof of this, due to S. Philipp, which I will not reveal (see pages 377, 378 of the paper…)

For the first time in my career, I am on sabbatical this semester, and will spend it at the Institute for Advanced Study (where there is a programme in analytic number theory this year).
Consequently, and also for the first time in a relatively long time, I am in the USA at the moment for professional reasons. This brings back memories of earlier times and rather pointless anthropological observations:

* On the down side, the Border bookshop “local” to where I stayed during the summer seems to have decided to dispense with a Science section. To replace it (?), there are two nice wide shelves of “Magic studies”. This is strange enough that I wonder if this is an isolated event (the equally local Barnes and Nobles still has a decent Science section).

* On the plus side, I would never have thought that I would buy Y. Meyer’s “Wavelets and operators” (in the Cambridge English edition) in a second hand bookstore located inside the Lakeview Museum in Peoria, Illinois, home base of the world’s largest scale model of the Solar System. By the way, if you intend to go see it — maybe as a pretext for bargain hunting maths books (the wavelet book cost 25 $, which is a fairly good deal certainly) –, be advised that the name Lakeview and the address West Lake Avenue are both misleading: the museum is quite far from the “lake” of Peoria, which in any case is just a slightly widened Illinois river; so do take good directions with you before leaving…

P.S. I checked: they don’t have a copy of “Galois Groups over Q“.

While cleaning up a bit the files on my laptop recently, I “discovered” an old computer program that I had written some time in 2003 or so, to display the graphical evolution of the partial sums of exponential sums, and more precisely of Kloosterman sums. In other words, given a prime number p and integers a and b, this program plots the points in the complex plane corresponding to the values

$latex S_k=\sum_{x=0}^{k}{\exp(2i\pi \frac{ax+b\bar{x}}{p})}$

for

$latex 0\leq k\leq p-1,$

and draws the line segments joining each successive S_k. (As usual, the bar over x indicates the inverse modulo p).

The resulting paths look fittingly psychedelic:

The rather badly written code (for moderately modern Linux systems) can be downloaded here; the compilation should be a straightforward

./configure ; make

the resulting executable kloostermania is then found in the src directory.

Alternatively, still for pretty recent Linux systems (at least, Fedora on Intel machines), you can get the executable here.

(I should probably state formally that the license is GPL, not that there’s much danger of a proprietary software company deciding to make a fortune and deprive mathematicians of much freedom to play with the code by selling derivatives of this program; of course, if anyone decides to add features or to make the program work, e.g., on Mac, this would be much appreciated…)

The program should be fairly easy to use; the File menu offers the possibility of saving the current picture as a PNG file or to print it to a PostScript file (the New and Open commands are just decoys). The Edit menu’s only interesting item is the Preference command, where the colors can be changed by clicking on the respective patches, and where the drawing of the axes can be disabled if desired. The View menu’s equally single item is used to enter the parameter for a new drawing; if the modulus p is not prime, the next prime will be selected (computed in a ridiculously inefficient way…) Finally, the current parameters and the final value of the sum are displayed at the bottom of the window…

I was yesterday at the Minnesota Science Museum; after admiring the nice probability machine

I found in the excellent museum store an even nicer coffee mug, which promises much refreshment and strenghtening of the mind in the future:

(These mugs are produced by the Unemployed Philosophers Guild).