E. Kowalski’s blog

Every mathematician who has ever exchanged (La)TeX files by email must have noticed lines starting

¿From…

appearing in the resulting dvi or pdf file.

These charmingly infuriating lines are due (if I understand things right) to TeX’s transforming the character “>” into the inverted question mark, and to the tendency of email programs to consider that a line starting with “From” means that an included email is starting, which must be quoted with “>”. Since mathematical papers tend to have sentences like “From this, it follows that…”, this is what we end up with, unless one is careful to regularly search the document for the telltale “>From” in order to remove the offending symbol (or one gets the reflex of cleverly writing “{}From” instead of “From”, something I just learnt from a coauthor.)

But I find it ironic that computers, which can apparently spell-check documents, correct their grammar, or attempt translating them into Esperanto, are unable to understand that sentences starting with “From” might be legitimate…

Kloostermania fans can look up what the new version 0.15 does on the Kloostermania page…

In particular, I’ve added the possibility to display Salié sums instead of Kloosterman sums. Precisely, it shows

instead of T(1,1;p) because the values of the latter are not as interesting, due to the fact that the root s of

modulo a prime are rather simple to compute….

This leads to an intriguing game to play: can you tell the difference between the two types of sums?

First, one must take a prime congruent to 1 modulo 4 (otherwise the Salié sum is zero in that case, which Kloosterman sums never are) for the question to be interesting. Then, there is a kind of theoretical/conjectural answer if you are allowed to look at many instances of the two sums (i.e., you can start the slideshow and observe it for a long time — skipping primes which are 3 mod 4 — without changing the type of sums): their distribution is not the same (conjecturally)! Precisely, the angles θ_p of the Salié sums, for primes which are 1 modulo 4, defined by

are equidistributed on [0,π] (for the Lebesgue measure; this is the wonderful theorem of Duke, Friedlander and Iwaniec), whereas one expects those of Kloosterman sums to be distributed according to the Sato-Tate measure

In particular, the Kloosterman sums should be more often “small”, in some sense, than the Salié sums since the density of the Sato-Tate measure vanishes at θ=0 (which corresponds to a maximal sum).

But what if you’re not allowed to start a long slideshow? For a fixed p, I don’t think one can expect to be able to guess more precisely, just from the values of the sums, than by throwing a coin and choosing Heads/Kloosterman, Tails/Salié. But I wonder if the shapes of the graph of partial sums (as drawn by the program…) could be used to extract more information to lead to a guess with better than even odds of being correct? Or if, at least, it could be used to shorten the length of time one would need to look at the slideshow before being sure of the answer from the distribution perspective?

This is just to indicate that the amazing Pocket Kloostermania program has been slightly updated. All users (if there are any except me…) are encouraged to upgrade…

It has been well said that Kloosterman sums are everywhere. Back in August, I showed how to visualize them on an ordinary laptop or desktop computer — software available also on its own page.

However, modern enlightened thought holds that availability on even the smallest of netbooks is not a good measure of ubiquity. Hence, without further ado, I am happy to introduce Pocket Kloostermania, the Android version. An installable package file can be downloaded from this link; it, and the source code, are also available now on the Kloostermania page; the license remains GPLv2.

Here’s what it looks like on the emulator.

Features/user manual are:

* The program has been installed and tested only on a Nexus One; it probably requires at least Android 2.0, but should be adaptable to earlier ones.

* On startup, the program displays the graph for S(1,1;173).

* Changes of orientation of the screen are recognized.

* Pressing the Menu soft key brings three choices (as seen on the picture):

Modulus and sum displays on the screen the modulus of the current sum (the two other parameters are always 1 and 1) as well as its value;

New sum shows a text input dialog where a new modulus can be entered; to finish input, press either Done on the soft keyboard, or the Back soft key, and the new sum will be drawn (with a modulus rounded to the nearest prime number);

About gives the version number and license information.

As pointed out by Roman Holowinsky, a Google search for “Sarnak” gives very amusing results… Apparently, this is the name of a character race on some video game (which I had never heard about); see (for instance) here for more, including pictures and many hilarious quotes…