From Euler’s Play with Infinite Series to the Anomalous Magnetic Moment

Abstract

During a first St. Petersburg period Leonhard Euler, in his early twenties, became interested in the Basel problem: summing the series of inverse squares. In the words of Andre Weil [W] “as with most questions that ever attracted his attention, he never abandoned it”. Euler introduced on the way the alternating “phi-series”, the better converging companion of the zeta function, the first example of a polylogarithm at a root of unity. He realized - empirically! - that odd zeta values appear to be new (transcendental?) numbers. It is amazing to see how, a quarter of a millennium later, the numbers Euler played with, “however repugnant” this game might have seemed to his contemporary lovers of the “higher kind of calculus”, reappeared in the first analytic calculation (by Laporta and Remiddi) of \(g-2\) - the anomalous magnetic moment of the electron, the most precisely calculated and measured physical quantity [K]. Mathematicians, on the other hand, are reviving the dream of Galois of uncovering a group structure of the periods, including the same multiple zeta values - the mixed Tate motives, inspired by ideas of Grothendieck and appearing in a variety of subjects - from algebraic geometry to Feynman amplitudes.