First edition, extremely rare offprints, of two of Ramanujan's most important papers. Two volumes, in the original printed paper wrappers.
 
A fine set.
 
According to Bruce Berndt (p. 52), On Certain Arithmetical Functions is "one of the most important papers in the history of number theory."  Ramanujan (1887-1920), the Indian mathematical genius who was discovered by the great Cambridge mathematician G.H. Hardy (1877-1947), is one of the most romantic figures in the history of mathematics and has been the subject of numerous books and even a movie, The Man Who Knew Infinity, starring Dev Patel as Ramanujan and Jeremy Irons as Hardy.  Ramanujan and Hardy "worked together for almost 5 years and Ramanujan s genius came to full flower. He was a much better mathematician than Hardy or Littlewood, or should I say, he was much more of a mathematical genius … Hardy later said that his greatest achievement in life was discovering Ramanujan.At one point, Hardy ranked all of the world s great mathematicians on a scale of 1 to 100, with 100 being the most brilliant. He gave himself a 25, the great David Hilbert an 80, and the 100 score was reserved for Ramanujan" (Linda Hall). The two offered papers "should be read together … They contain, in particular, very original and important contributions to the theory of the representation of numbers by sums of squares" (Papers, p. xxxiv).  The history of this theory "may be traced back to Diophantus but begins effectively with Girard's (or Fermat's) theorem that a prime 4m +1 is the sum of two squares. Almost every arithmetician of note since Fermat has contributed to the solution of the problem, and it has its puzzles for us still" (Hardy, p. 132).  "Ramanujan's contributions to the subject are set out in two substantial papers in the Transactions of the Cambridge Philosophical Society [the offered papers] … The papers are highly original: they are characteristic of Ramanujan at his best. They contain many remarkable theorems which are undeniably new, and conjectures still more remarkable, which were confirmed later by Mordell; and the general level of the analysis is astonishingly high. In particular, the second paper contains all the formal theory of Ramanujan's sum " (ibid., pp. 136-7).  In the first paper Ramanujan introduces his tau function, which appears in a kind of error term involved in counting the number of ways of expressing an integer as a sum of 24 squares. Based on some computations in special cases, Ramanujan conjectured that the tau function had several remarkable properties. Some of these were proved by Louis J. Mordell soon after Ramanujan s paper was published, but one of them, which became known as the tau conjecture, resisted proof until 1974. Recently, the tau function has found applications in string theory.  The Ramanujan sum introduced in the second paper was used by him to derive many important infinite series expansions for arithmetic functions in number theory. Curiously, it also has applications in signal-processing.  No copies on OCLC or RBH.