Eulerian Polynomials and Numbers, Bernoulli Numbers, and the Riemann Zeta-Function

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.