By Pierre Cartier (auth.), Michel Waldschmidt, Pierre Moussa, Jean-Marc Luck, Claude Itzykson (eds.)

The current e-book comprises fourteen expository contributions on numerous themes attached to quantity conception, or Arithmetics, and its relationships to Theoreti cal Physics. the 1st half is mathematically orientated; it offers typically with ellip tic curves, modular kinds, zeta capabilities, Galois thought, Riemann surfaces, and p-adic research. the second one half reviews on issues with extra direct actual curiosity, reminiscent of periodic and quasiperiodic lattices, or classical and quantum dynamical structures. The contribution of every writer represents a brief self-contained path on a particular topic. With only a few necessities, the reader is out there a didactic exposition, which follows the author's unique viewpoints, and sometimes incorpo premiums the latest advancements. As we will clarify under, there are powerful relationships among the several chapters, even supposing each contri bution will be learn independently of the others. This quantity originates in a gathering entitled quantity idea and Physics, which happened on the Centre de body, Les Houches (Haute-Savoie, France), on March 7 - sixteen, 1989. the purpose of this interdisciplinary assembly was once to collect physicists and mathematicians, and to offer to contributors of either com munities the possibility of changing rules, and to learn from every one other's particular wisdom, within the region of quantity conception, and of its functions to the actual sciences. Physicists were given, more often than not in the course of the software of lectures, an exposition of a few of the elemental tools and result of Num ber thought that are the main actively utilized in their branch.

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An Introduction to Zeta Functions number-theoretic point of view, it is an ideal of Z[i], that is a subgroup of Z[i] (for the addition), stable under multiplication by any element in Z[i]. The set of multiples of z (including 0) is the principal ideal generated by z, to be denoted by (z). For instance, the ideal (0) consists of 0 only, it is called the zero ideal. Let us state the main properties of ideals in Z[i] : a) Every nonzero ideal in Z[i] is a principal ideal. Namely, let I be such an ideal.

5. Sums of squares Fermat considered the following problem: Represent, if possible, an integer n 2': 1 as a sum of two squares n = a2 + b2 • It amounts to represent n as the norm of some Gaussian integer tv = a + bi. 3, we can write tv as a product Utvl ... 15) a 2 + b2 = N(a + bi) = N(tvd'" N(tvN). 4, the norm of a Gaussian prime tv j is equal to 2, to a prime number p == 1 mod. 4, or to the square of a prime number p == 3 mod. 4. The following criterion, due to Fermat, follows immediately: An integer n 2': 1 is a sum of two squares if and only if every prime divisor of n congruent to 3 mod.

0 Proof. Uniqueness: Suppose given two decompositions z = UWI·· ·WN , z = u'w~·· ·wN,. Without loss of generality, assume N :::; N'. Since the Gaussian prime w~ divides z = UWI ... WN and does not divide the unit u, it divides one of the factors WI, ... , W N (lemma 2). For instance assume w~ divides WI. Since both WI, W~ are normalized Gaussian primes, this implies WI = W~. We are done if N' = 1. Otherwise after simplifying we get and continuing the previous argument, we may assume that, after a permutation of factors if necessary, we have If N = N', we derive U = u' from these equalities and we are done.