This reviewer can do no better than reproduce portions of the authors’ Introduction, which is absolutely superb!
From the introduction: “The centerpiece of this volume is the partition function \(p(n)\). Featured in this book are congruences for \(p(n)\), ranks and cranks of partitions, the Ramanujan \(\tau\)-function, the Rogers-Ramanujan functions, and the unpublished portion of Ramanujan’s paper on highly composite numbers [Proc. Lond. Math. Soc. (2) 14, 347–409 (1915; JFM 45.1248.01, JFM 45.0286.02)].
“The first three chapters are devoted to ranks and cranks of partitions. In 1944, F. J. Dyson [Eureka 8, 10–15 (1944)] defined the rank of a partition to be the largest part minus the number of parts. If \(N(m,t,n)\) denotes the number of partitions of \(n\) with rank congruent to \(m\) modulo \(t\), then Dyson conjectured that
\[ N(k,5,5n+4)=\frac{p(5n+4)}{5},\quad 0\leq k\leq 4,\tag{1.0.1} \]
and
\[ N(k,7,7n+5)=\frac{p(7n+5)}{7},\quad 0\leq k\leq 6\tag{1.0.2} \]
Thus, if (1.0.1) and (1.0.2) were true, the partitions counted by \(p(5n+4)\) and \(p(7n+5)\) would fall into five and seven equinumerous classes, respectively, thus providing combinatorial explanations and proofs for Ramanujan’s famous congruences \(p(5n + 4)\equiv 0 \pmod {5}\) and \(p(7n+5)\equiv 0\pmod {7}\). Dyson’s conjectures were first proved in 1954 by A. O. L. Atkin and H. P. F. Swinnerton-Dyer [Proc. Lond. Math. Soc. (3) 4, 84–106 (1954; Zbl 0055.03805)].
“Dyson observed that the corresponding analogue to (1.0.1) and (1.0.2) does not hold for the third famous Ramanujan congruence \(p(11n+6)\equiv 0 \pmod {11}\), and so he conjectured the existence of a statistic that he called the crank that would combinatorially explain this congruence. In his doctoral dissertation [Generalizations of Dyson’s rank, Ph.D. Thesis, Pennsylvania State University, University Park, PA (1986)], F. G. Garvan defined a crank for vector partitions, which became the forerunner of the true crank, which was discovered by G. E. Andrews and F. G. Garvan [Bull. Am. Math. Soc., New Ser. 18, No. 2, 167–171 (1988; Zbl 0646.10008)] […]
“Although Ramanujan did not record any written text about ranks and cranks in his lost notebook [The Lost Notebook and other unpublished papers. Berlin: Springer-Verlag; New Delhi: Narosa Publishing House (1988; Zbl 0639.01023)], he did record theorems about their generating functions. Chapter 2 is devoted to the five and seven-dissections of each of these two generating functions. Cranks are the exclusive topic of Chapter 3, where dissections for the generating function for cranks are studied, but now in the context of congruences. A particular formula found in the lost notebook and proved in Chapter 4 is employed in our proofs in Chapter 3. […]
“G. H. Hardy (ed.) [Math. Z. 9, 147–153 (1921; JFM 48.0150.02)] extracted a portion of Ramanujan’s manuscript and added several details in giving proofs of his aforementioned famous congruences for the partition function, namely, \[ p(5n+4)\equiv 0 \pmod {5}, \;p(7n+5)\equiv 0 \pmod {7},\;p(11n+6)\equiv 0\pmod {11}.\tag{1.0.3} \]
[…]
“These congruences (1.0.3) are the first cases of the infinite families of congruences
\[ p(5^kn+\delta_{5,k})\equiv 0 \pmod {5^k},\tag{1.0.4} \]
\[ p(7^kn+\delta_{7,k})\equiv 0 \pmod {7^{[k/2]+1}},\tag{1.0.5} \]
\[ p(11^kn+\delta_{11,k})\equiv 0 \pmod {11^k}, \] where \(\delta_{p,k}:\equiv 1/24 \pmod{p^k}\). In Ramanujan’s manuscript, he actually gives a complete proof of (1.0.4), but many of the details are omitted. These details were supplied by G. N. Watson [J. Reine Angew. Math. 179, 97–128 (1938; Zbl 0019.15302, JFM 64.0122.02)] […]
“Since proofs of (1.0.4) and (1.0.5) can now be found in several sources (which we relate in Chapter 5), there is no need to give proofs here. […] It was surprising for us to learn that Ramanujan had also found congruences for \(p(n)\) for the moduli 13, 17, 19, and 23 and had formulated a general conjecture about congruences for any prime modulus. However, unlike (1.0.3), these congruences do not give divisibility of \(p(n)\) in any arithmetic progressions. In his doctoral dissertation, J. M. Rushforth [Congruence properties of the partition function and associated functions, Doctoral Thesis, University of Birmingham (1950), Proc. Camb. Philos. Soc. 48, 402–413 (1952; Zbl 0047.04302)] supplied all of the missing details for Ramanujan’s congruences modulo 13, 17, 19, and 23. Since Rushforth’s work has never been published and since his proofs are motivated by those found by Ramanujan, we have decided to publish them here for the first time. In fact, almost all of Rushforth’s thesis is devoted to Ramanujan’s unpublished manuscript on \(p(n)\) and \(\tau(n)\), and so we have extracted from it further proofs of results claimed by Ramanujan in this famous manuscript.
“Ramanujan’s general conjecture on congruences for prime moduli was independently corrected, proved, and generalized in two distinct directions by H. H. Chan and J.-P. Serre and by S. Ahlgren and M. Boylan [Invent. Math. 153, No. 3, 487–502 (2003; Zbl 1038.11067)]. The proof by Chan and Serre is given here for the first time. […]
“Chapter 6 is devoted to six entries on page 189 of the lost notebook [loc. cit.], all of which are related to the content of Chapter 5, and to entries on page 182, which are related to Ramanujan’s paper on congruences for \(p(n)\) [Proc. Camb. Philos. Soc. 19, 207–210 (1919; JFM 47.0885.01)] and of course also to Chapter 5. In particular, we give proofs of two of Ramanujan’s most famous identities, immediately yielding the first two congruences in (1.0.3). On page 182, we also see that Ramanujan briefly examined congruences for \(p_r(n)\), where \(p_r(n)\) is defined by
\[ (q;q)^r_\infty=\sum_{n=0}^\infty p_r(n)q^n,\quad |q|<1. \]
Apparently, page 182 is page 5 from a manuscript, but unfortunately all of the remaining pages of this manuscript are likely lost forever. We have decided also to discuss in Chapter 6 various scattered, miscellaneous entries on \(p(n)\). Most of this mélange can be found in Ramanujan’s famous paper with Hardy establishing their asymptotic series for \(p(n)\) [Proc. Lond. Math. Soc. (2) 17, 75–115 (1917; JFM 46.0198.04)].
“In Chapter 7, we examine nine congruences that make up page 178 in the lost notebook. These congruences are on generalized tau functions and are in the spirit of Ramanujan’s famous congruences for \(\tau(n)\) discussed in Chapter 5.
“The Rogers-Ramanujan functions are the focus of Chapter 8, wherein Ramanujan’s 40 famous identities for these functions are examined. Having been sent some, or possibly all, of the 40 identities in a letter from Ramanujan, L. J. Rogers [Proc. Lond. Math. Soc. (2) 19, 387–397 (1921; JFM 48.0151.02)] proved eight of them, with G. N. Watson [J. Indian Math. Soc. 20, 57–69 (1934; Zbl 0009.33707, JFM 60.0184.01)] later providing proofs for six further identities as well as giving different proofs of two of the identities proved by Rogers. For several years after Ramanujan’s death, the list of 40 identities was in the hands of Watson, who made a handwritten copy for himself, and it is this copy that is published in [The Lost Notebook and Other Unpublished Papers (loc. cit.)]. Fortunately, he did not discard the list in Ramanujan’s handwriting, which now resides in the library at Trinity College, Cambridge. Approximately ten years after Watson’s death, B. J. Birch [A look back at Ramanujan’s notebooks. Math. Proc. Camb. Philos. Soc. 78, 73–79 (1975; Zbl 0305.10002)] found Watson’s copy in the library at Oxford University and published it in 1975, thus bringing it to the mathematical public for the first time. D. Bressoud [Proof and generalization of certain identities conjectured by Ramanujan, Ph.D. Thesis, Temple University (1977)] and A. J. F. Biagioli [Glasg. Math. J. 31, No. 3, 271–295 (1989; Zbl 0679.10018)] subsequently proved several further identities from the list.
“Our account of the 40 identities in Chapter 8 is primarily taken from a Memoir [Mem. Am. Math. Soc. 880, 96 p. (2007; Zbl 1118.11044) by B. C. Berndt, G. Choi, Y.-S. Choi, H. Hahn, B. P. Yeap, A. J. Yee, H. Yesilyurt and J. Yi. The goal of these authors was to provide proofs for as many of these identities as possible that were in the spirit of Ramanujan’s mathematics. In doing so, they borrowed some proofs from Rogers, Watson, and Bressoud, while supplying many new proofs as well. After the publication of this Memoir in which proofs of 35 of the 40 identities were given in the spirit of Ramanujan, H. Yesilyurt [J. Number Theory 129, No. 6, 1256–1271 (2009; Zbl 1219.11063), J. Math. Anal. Appl. 388, No. 1, 420–434 (2012; Zbl 1272.11037)] devised ingenious and difficult proofs of the remaining five identities, and so these papers are the second primary source on which Chapter 8 is constructed.
“Chapter 9 is devoted to one general theorem on certain sums of positive integral powers of theta functions, and five examples in illustration. […]
“In 1915, the London Mathematical Society published Ramanujan’s paper on highly composite numbers [loc. cit.], [Collected Papers, 78–128]. However, this is only part of the paper that Ramanujan submitted. The London Mathematical Society was in poor financial condition at that time, and to diminish expenses, they did not publish all of Ramanujan’s paper. Fortunately, the remainder of the paper has not been lost and resides in the library at Trinity College, Cambridge. In its original handwritten form, it was photocopied along with Ramanujan’s lost notebook in 1988 [loc. cit.]. J.-L. Nicolas and G. Robin prepared an annotated version of the paper for the first volume of the Ramanujan Journal in 1997 [Ramanujan J. 1, No. 2, 119–153 (1997; Zbl 0917.11043)]. In particular, they inserted text where gaps occurred, and at the end of the paper, they provided extensive commentary on research in the field of highly composite numbers accomplished since the publication of Ramanujan’s original paper. Chapter 10 contains this previously unpublished manuscript of Ramanujan on highly composite numbers, as completed by Nicolas and Robin, and a moderately revised and extended version of the commentary originally written by Nicolas and Robin.”
Reviewer’s remark: The Introduction includes wonderful historical context for Ramanujan’s partition work; in the same vein, the first section of Chapter 5 is not to be missed!
For the two previous volumes see Zbl 1075.11001 and Zbl 1180.11001.