## Stickelberger’s copy of Jacobi’s “Canon arithmeticus”

I am currently the head of the Mathematics Library at ETH (which is separate from the main library). A few days ago, I surveyed some of the (relatively) old books in our collection with one of the librarians, just to see if some of these should be handled in a special way. We didn’t find anything really out of the ordinary (no copy of Poincaré’s works heavily annotated by H. Weyl, I’m afraid), but one book has some historical interest: it is (or seems to be) Stickelberger’s copy of Jacobi’s “Canon arithmeticus”

a table of primitive roots and discrete logarithms for primes up to 1000.

Stickelberger’s signature is found on one of the first pages

The table itself, as it took me a few minutes to understand (my Latin being non-existent), lists for each prime $p\leq 1000$ the “Numeri” $1\leq N\leq p-1$ and the “Index” $1\leq I\leq p-1$, which are defined by the relation
$N=\rho^I,$
for some primitive root $\rho$ modulo $p$, which can be identified easily by looking for the number for which $I$ is equal to $1$:
$\rho=\rho^1.$

So above we see that Jacobi selected $2$ as primitive root modulo $5$ and $11$, and $3$ as primitive root modulo $7$, or $6$ as primitive root modulo $13$. Obligingly, he also indicates the factorization of $p-1$ (so that all primitive roots can be easily found by checking whether the corresponding index is coprime with $p-1$).

Like the copy which was digitized by Google, Stickelberger’s has a list of corrections at the end, and most (if not all: I didn’t check…) of these are incorporated in pencil in the main text, as here with $p=71$:

However, Stickelberger (if it was him) also had another list of corrections, written down on a separate loose sheet of paper inserted at the end of the book.

These corrections are reproduced from the paper On quasi-mersennian numbers by Lieutenant Colonel Allan Cunningham in Vol. 41 of the Messenger of Mathematics (a volume which seems famous in statistical circles because it contains, ten pages later, an important paper of R.A. Fisher on maximum likelihood…) But even Cunningham’s corrections contain a few mistakes, which Stickelberger reports (though with question marks):

Indeed, for $p=757$, the primitive root chosen by Jacobi is $\rho=2$ and we have
$2^{468}=565\bmod{757},$
instead of $568$ reported by Cunningham (and $168$ in the Canon).

As far as I could see during my quick inspection, there are no further annotations or comments by Stickelberger, nor any date indicating when he acquired this book. The publication date is 1839, and the only other indication is that the volume of the Messenger of Mathematics with Cunningham’s paper appeared in 1912. I also do not known when and how the book entered the collection of ETH.

## Upcoming events!

There will be quite a few number-theoretic activities that I will be involved-in this year. In chronological order:

(1) In March, on Friday the 15th and Saturday the 16th, there will first be the 10th edition of the ETH-EPFL Number Theory Days, this time in Lausanne. The web page is currently not very informative (except for links to the previous editions…), but the speakers this year will be L. Berger (ENS Lyon), E. Lindenstrauss (Hebrew University), H. Oh (Brown University), N. Templier (Princeton University) and J. Wolf (École Polytechnique). This is organized by Ph. Michel and myself.

(2) Immediately following, there will be a conference on “Equidistribution in number theory and dynamics” at the Forschungsinstitut für Mathematik of ETH, organized jointly by M. Einsiedler, E. Lindenstrauss, Ph. Michel and myself, from March 18 to 22. There is a web page with the current list of speakers. We would especially like to invite young mathematicians to apply here for financial support if they wish to attend this conference (the deadline indicated is January 15th, but a few more days should not hurt).

(3) During the first week of June, again at FIM, G. Wustholz and myself are organizing a conference to celebrate the 25th anniversary of the Number Theory Seminar at ETH. Additional details will appear soon….

(4) Finally, from June 17 to 21, in sunny Marseille, R. de la Bretèche, Ph. Michel, J. Rivat and myself are co-organizing a conference on Analytic Number Theory in honor of É. Fouvry’s 60th birthday. This will be held at CIRM, which is a also a very nice place indeed to do mathematics. The web page for the conference is here; registration to the conference should be done on the CIRM website (the registration form is not there yet).

## The rights of frogs of toads

How more considerate of our amphibian cousins can you get?

(Translation: “Amphibian migration / Obere Geerenstrasse / From the beginning of Mars to the end of Mars / 18:00 afternoon to 06:30 morning / Inaccessible”)

For an additional Swiss note, the next panel, which unfortunately I couldn’t photograph in the same frame, was a reminder that their is a votation this Sunday.

## Number Theory Days 2012

Since 2005, the number theorists at ETH Zürich and at EPF Lausanne organize every year the Number Theory Days, which present five talks, in all areas of number theory, over two days. (Actually, I’ve seen one web page claim that the tradition started in 2004, but I didn’t find any other reference for it…)

The 2012 edition will be held in Zürich on March 30 and 31, co-organized by Philippe Michel and myself, with the support of the FIM. Here is the official web page and the beautiful poster:

We especially encourage your researchers interested in attending (PhD students, postdocs, in particular) to write to the FIM coordinator, as indicated on the web page, to register, and to indicate if they wish to request funding for travel and local expenses.

## Birds of the Léman

My last post about birds concerned the great crested grebe (which appears in a most hilarious way in Scoop — “Lord Copper,” he was saying, “no man shall call me a liar unchastised. The great crested grebe does hibernate.”)

Although a more recent week-end excursion on the Léman was richer in raptors, it was also a pleasure to watch the noble heron (Héron cendré, I guess) guarding the piers:

However, the thrilling part was to watch raptors fishing; to be precise, I think most of them were the same species

which — interpreting my bird-book — is quite likely to be the Milan noir (Milvus migrans) (it might be the Milan royal, but that seems much more unlikely). Of course, most of my attempts to photograph the fishing of the milans resulted in blurry pictures of the surface of the lake; however, some succeeded, like this one

or that one