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By Pierre E. Cartier, Bernard Julia, Pierre Moussa, Pierre Vanhove

The relation among arithmetic and physics has an extended heritage, during which the function of quantity thought and of alternative extra summary components of arithmetic has lately develop into extra prominent.More than ten years after a primary assembly in 1989 among quantity theorists and physicists on the Centre de body des Houches, a moment 2-week occasion keen on the wider interface of quantity conception, geometry, and physics.This e-book is the results of that fascinating assembly, and collects, in 2 volumes, prolonged types of the lecture classes, by means of shorter texts on unique issues, of eminent mathematicians and physicists.

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Example text

Moreover, the new function 2 L(x) := L(x) (mod π 2 /2) from P1 (R) to R/ π2 Z, just like its complex analogue D(z), satisfies “clean” functional equations with no logarithm terms, in particular the reflection properties L(x) + L(1 − x) = L(1), L(x) + L(1/x) = −L(1) and the 5-term functional equation L(x) + L(y) + L 1−x 1−y + L(1 − xy) + L 1 − xy 1 − xy = 0. If we replace L by L in the left-hand sides of these three equations, then their right-hand sides must be replaced by piecewise continuous functions whose values depend on the positions of the arguments: L(1) in the first equation, 2L(1) for x > 0 or −L(1) for x < 0 in the second, and −3L(1) for x, y < 0, xy > 1 and +3L(1) otherwise in the third.

We discuss only some elementary aspects concerning the asymptotic properties of Nahm’s q-expansions, referring the reader for the deeper parts of the theory, concerning the (in general conjectural) interpretation of these qseries as characters of rational conformal field theories, to the beautiful article by Nahm in this volume. As well as the two original footnotes to Chapter I, which are indicated by numbers in the text and placed at the bottom of the page in the traditional manner, there are also some further footnotes, indicated by boxed capital letters in the margin and placed at the end of the chapter, which give updates or comments on the text of the older article or else refer the reader to the sections of Chapter II where the topic in question is developed further.

E. The p-adic dilogarithm and the dianalog. The next dilogarithm variant we mention is the p-adic dilogarithm, studied by R. Coleman and other authors. We fix a prime number p and define (p) Li2 (x) = n>0, p∤n xn . n2 (14) This function can be written as Li2 (x)−p−2 Li2 (xp ), so in the complex domain it is simply a combination of ordinary dilogarithms and of no independent interest, but because we have omitted the terms in (14) with p’s in the denominator, the power series converges p-adically for all p-adic numbers x with (p) valuation |x|p < 1.

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