Ramanujan’s Notebooks

Srinivasa Ramanujan occupies a central but singular position in mathematical history. The pathway to enduring, meaningful, creative mathematical research is by no means the same for any two individuals, but for Ramanujan, his path was strewn with the impediments of poverty, a lack of a university education, the absence of books and journals, working in isolation in his most creative years, and an early death at the age of 32. Few, if any, of his mathematical peers had to encounter so many formidable obstacles.

The origin of Chapter 1 probably is found in Ramanujan’s early school days and is therefore much earlier than the remainder of the notebooks. Rules for constructing certain rectangular arrays of natural numbers are given. Most of Ramanujan’s attention is devoted to constructing magic squares. A magic square is a square array of (usually distinct) natural numbers so that the sum of the numbers in each row, column, or diagonal is the same. In some instances, the requirement on the two diagonal sums is dropped. In the notebooks, Ramanujan uses the word “corner” for “diagonal.” We emphasize that the theory of magic squares is barely begun by Ramanujan in Chapter 1. Considerably more extensive developments are contained in the books of W. S. Andrews [1] and Stark [1], for example.

Chapter 2 is fairly elementary, but several of the formulas are very intriguing and evince Ramanujan’s ingenuity and cleverness. Ramanujan gives more proofs in this chapter than in most of the later chapters.

Although no combinatorial problems are mentioned in Chapter 3, much of the content of this chapter belongs under the umbrella of combinatorial analysis. Another primary theme in Chapter 3 revolves around series expansions of various types. However, the deepest and most interesting result in Chapter 3 is Entry 10, which separates the two main themes but which has some connections with the former. Entry 10 offers a highly general and potentially very useful asymptotic expansion for a large class of power series. As with Chapter 2, Ramanujan very briefly sketches the proofs of some of his findings, including Entry 10.

The first seven paragraphs of Chapter 4 are concerned with iterated exponential functions and constitute a sequel to a large portion of Chapter 3 wherein the Bell numbers, single-variable Bell polynomials, and related topics are studied. Recall that the Bell numbers B(n), 0 ≤ n ≤ ∞, may be defined by They were first thoroughly studied in print by Bell [1], [2] approximately 25–30 years after Ramanujan had derived several of their properties in the notebooks. Further iterations of the exponential function appear to have been scarcely studied in the literature. The most extensive study was undertaken by Bell [2] in 1938. Becker and Riordan [1] and Carlitz [1] have established arithmetical properties for these generalizations of Bell numbers. Also, Ginsburg [1] has briefly considered such iterates. For a combinatorial interpretation of numbers generated by iterated exponential functions, see Stanley’s article [1, Theorem 6.1].

Chapter 5 contains more number theory than any of the remaining 20 chapters. Of the 94 formulas or statements of theorems in Chapter 5, the great majority pertain to Bernoulli numbers, Euler numbers, Eulerian polynomials and numbers, and the Riemann zeta-function. As is to be expected, most of these results are not new. The geneses of Ramanujan’s first published paper [4] (on Bernoulli numbers) and fourth published paper [7] (on sums connected with the Riemann zeta-function) are found in Chapter 5. Most of Ramanujan’s discoveries about Bernoulli numbers that are recorded here may be found in standard texts, such as those by Bromwich [1], Nielsen [5], Nörlund [2], and Uspensky and Heaslet [1], for example.

In a letter written to A. Holmboe on January 16, 1826, Abel [3] declared that “Divergent series are in general deadly, and it is shameful that anyone dare to base any proof on them.” This admonition would have been vehemently debated by Ramanujan. Much like Euler, Ramanujan employed divergent series in a variety of ways to establish a diversity of results, most of them valid but a few not so. Divergent series are copious throughout Ramanujan’s notebooks, but especially in Chapter 6 of the second notebook, or in Chapter 8 of the first notebook. Since Ramanujan always uses equality signs in stating identities that involve one or more divergent series, one might be led to believe that Ramanujan probably made no distinction between convergent and divergent series. However, the occasional discourse in Chapter 6 is firm evidence that Ramanujan made such a distinction.

The first 14 sections of Chapter 8 comprise but 41/2 of the 12 pages in this chapter. Initial results are concerned with partial sums of the harmonic series and the logarithmic derivative ψ(x) of the gamma function. As might be expected, most of these results are very familiar. Ramanujan actually does not express his formulas in terms of ψ(x) but instead in terms of \(\varphi \left( x \right) = \sum\nolimits_{k = 1}^x {1/k.} \). As in Chapter 6, Ramanujan really intends ϕ(x) to be interpreted as ϕ(x + 1) + γ, for all real x, where γ denotes Euler’s constant. These 14 sections also contain several evaluations of elementary integrals of rational functions. Certain of these integrals are connected with an interesting series \(\sum\nolimits_{k = 1}^\infty {1/\left\{ {{{\left( {kx} \right)}^3} - kx} \right\},} \), which Ramanujan also examined in Chapter 2.