THE RAMANUJAN JOURNAL, Vol. 6, No. 1 (2002), 5-6.
The study of sums of squares is one of the oldest and most revered in mathematics. In chapter 11 of his book, Great Moments in Mathematics Before 1650, Howard Eves says "Now in the history of mathematics there is one man who stands out as probably the first true genius in the field of number theory, and one whose works so profoundly influenced later European number theorists that the production of this work can well be labeled a GREAT MOMENT IN MATHEMATICS. The man is Diophantus of Alexandria, and the work alluded to is his famous Arithmetica."
This book posed numerous problems whose solutions were to be integers. Such problems have become known as Diophantine problems. Probably the most famous of all Diophantus' problems is Problem 8 of Book II of the Arithmetica which reads "To divide a given square number into two squares." As we are all aware, adjoining this problem in the margin of his copy of Bachet's translation, Fermat wrote his famous "Last Theorem" with the tantalizing comment that he "did indeed have a proof . . . but the margin is too narrow to contain it."
In Book IV of the Arithmetica, Diophantus considers problems involving sums of four squares, and Bachet asserted that Diophantus assumed that every number was either a square or the sum of two, three or four squares. Euler, Fermat and others attempted to prove this assertion, and in the middle of the eighteenth century, Lagrange succeeded. From then until today discoveries about sums of squares have been of great significance in mathematics.
One of the milestones in the work on sums of squares was Jacobi's Fundamenta Nova Theoriae Functionum Ellipticarum in 1829. In this book, Jacobi developed the method of elliptic functions sufficiently to produce exact formulae for rs(n), the number of representations of n as a sum of s squares when s = 2, 4, 6 or 8.
At the beginning of the twentieth century, Mordell pioneered the application of the theory of modular forms to sums of squares problems. This approach was sufficiently powerful that it came to dominate much of the work in this area during the twentieth century.
In the mid-1980s, Grosswald published his book Representations of Integers as the Sums of Squares. In chapters 8 and 9 of this book, Grosswald illuminated the nineteenth century elliptic function methods of Jacobi, Glaisher and others. In chapter 10, he presented the modular form ideas of Mordell and others that flourished in the twentieth century.
Here at the beginning of the twenty-first century, we welcome Steve Milne's extensive contribution to the elliptic function tradition in the study of sums of squares. The genesis of Milne's studies lies in his decades-long development of generalized hypergeometric and q-hypergeometric series related to the classical groups. Subsequently he saw how to amalgamate those studies with his combinatorial insights and classical elliptic function theory to obtain the powerful results in this monograph. The original elliptic function methods provided formulas for rs(n) when s was small (usually <24). For example, examination of Jacobi's Fundamenta Nova, sections 40-42, reveals twenty plus identities related to generating functions for sums of squares. Milne produces infinite families of identities wherein the aforementioned identities of Jacobi are each the first member of one family. In addition, he proves the conjectured identities of Kac and Wakimoto, which involve triangular numbers and which arose in the study of Lie algebras. (It should be noted that Don Zagier has independently proved these conjectures utilizing the theory of modular forms.)
This impressive paper will undoubtedly spur others both in elliptic functions and in modular forms to build on these wonderful discoveries.
George E. Andrews
