Euler is one of the greatest and most prolific mathematicians of
all time. He wrote the first accessible books on calculus, created
the theory of circular functions, and discovered new areas of
research such as elliptic integrals, the calculus of variations,
graph theory, divergent series, and so on. It took hundreds of
years for his successors to develop in full the theories he began,
and some of his themes are still at the center of today's
mathematics. It is of great interest therefore to examine his work
and its relation to current mathematics. This book attempts to do
that. In number theory the discoveries he made empirically would
require for their eventual understanding such sophisticated
developments as the reciprocity laws and class field theory.His
pioneering work on elliptic integrals is the precursor of the
modern theory of abelian functions and abelian integrals. His
evaluation of zeta and multizeta values is not only a fantastic and
exciting story but very relevant to us, because they are at the
confluence of much research in algebraic geometry and number theory
today (Chapters 2 and 3 of the book). Anticipating his successors
by more than a century, Euler created a theory of summation of
series that do not converge in the traditional manner.Chapter 5 of
the book treats the progression of ideas regarding divergent series
from Euler to many parts of modern analysis and quantum physics.
The last chapter contains a brief treatment of Euler products.
Euler discovered the product formula over the primes for the zeta
function as well as for a small number of what are now called
Dirichlet $L$-functions. Here the book goes into the development of
the theory of such Euler products and the role they play in number
theory, thus offering the reader a glimpse of current developments
(the Langlands program). For other wonderful titles written by this
author see: ""Supersymmetry for Mathematicians: An Introduction"",
""The Mathematical Legacy of Harish-Chandra: A Celebration of
Representation Theory and Harmonic Analysis"", ""The Selected Works
of V.S. Varadarajan"", and ""Algebra in Ancient and Modern Times"".
General
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