From the authors’ superb and very rewarding introduction, (Chapter 1): “Number theory and classical analysis are in the spotlight in the present book, which is the fourth of five projected volumes, wherein the authors plan to discuss all the claims made by S. Ramanujan in [(*) “The Lost Notebook and other unpublished papers.” Berlin: Springer-Verlag (1988; Zbl 0639.01023)]. As in our previous volumes, and in the sequel, we liberally interpret lost notebook not only to include the original lost notebook found by the first author in the library at Trinity College, Cambridge, in March 1976, but also to include all of the material published in (*). This includes letters that Ramanujan wrote to G. H. Hardy from nursing homes, several partial manuscripts, and miscellaneous papers. Some of these manuscripts are located at Oxford University, are in the handwriting of G. N. Watson, and are “copied from loose papers.” However, it should be emphasized that the original manuscripts in Ramanujan’s handwriting can be found at Trinity College Library; Cambridge.”
“Chapter 2 is devoted to two intriguing identities involving double series of Bessel functions found on page 335 of (*). One is connected with the classical circle problem, while the other is conjoined to the equally famous Dirichlet divisor problem. The double series converge very slowly, and the identities were extremely difficult to prove. Initially, the second author and his collaborators, S. Kim and A. Zaharescu, were not able to prove the identities with the order of summation as prescribed by Ramanujan, i.e., the identities were proved with the order of summation reversed [Adv. Math. 229, No. 3, 2055–2097 (2012; Zbl 1236.33010); Math. Ann. 335, No. 2, 249–283 (2006; Zbl 1100.33001)]. It is possible that Ramanujan intended that the summation indices should tend to infinity “together”. The three authors therefore also proved the two identities with the product of the summation indices tending to \(\infty\) (*). Finally, these authors proved Ramanujan’s first identity with the order of summation as prescribed by Ramanujan [Adv. Math. 236, 24–59 (2013; Zbl 1326.11059)].
In Chap. 2, we provide proofs of the two identities with the order of summation indicated by Ramanujan in the first identity and with the order of summation reversed in the second identity. We also establish the identities when the product of the two indices of summation tends to infinity.”
“It came as a huge surprise to us while examining pages in “The Lost Notebook and other unpublished papers” when we espied famous formulas of N. S. Koshliakov and A. P. Guinand, although Ramanujan wrote them in slightly disguised forms. Moreover, we discovered that Ramanujan had found some consequences of these formulas that had not theretofore been found by any other authors. We are grateful to Y. Lee and J. Sohn for their collaboration on these formulas, which are the focus of Chap.3.
Chapter 4, on the classical gamma function, features two sets of claims. We begin the chapter with some integrals involving the gamma function in the integrands. Secondly, We examine a claim that reverts to a problem (*) that Ramanujan submitted to the Journal of the Indian Mathematical Society, which was never completely solved. On page 339 in (*), Ramanujan offers a refinement of this problem, which was proved by the combined efforts of E. Karatsuba [J. Comput. Appl. Math. 135, No. 2, 225–240 (2001; Zbl 0988.33001)] and H. Alzer [Bull. Lond. Math. Soc. 35, No. 5, 601–607 (2003; Zbl 1027.33002)].
Hypergeometric functions are featured in Chap. 5. This chapter contains two particularly interesting results. The first is an explicit representation for a quotient of two particular bilateral hypergeometric series.”
“Ramanujan’s formula is so unexpected that no one but Ramanujan could have discovered it! The second is a beautiful continued fraction, for which S. Y. Kang, S. G. Lim, and J. Sohn [J. Math. Anal. Appl. 307, No. 1, 153–166 (2005; Zbl 1068.33018)] found two entirely different proofs, each providing a different understanding of the entry.
Chapter 6 contains accounts of two incomplete manuscripts on Euler’s constant \(\gamma\).
S. Kim kindly collaborated with the second author on Chap. 7, on an unusual problem examined in a rough manuscript by Ramanujan on Diophantine approximation [Ramanujan J. 31, No. 1-2, 83–95 (2013; Zbl 1329.11073)]. This manuscript was another huge surprise to us, for it had never been noticed by anyone, to the best of our knowledge, that Ramanujan had derived the best possible Diophantine approximation to \(e^{2/a}\), which was first proved in print approximately 60 years after Ramanujan had found his proof.”
“At the beginning of Chap. 8, in Sect. 8.1, we relate that Ramanujan had anticipated the famous work of L. G. Sathe [J. Indian Math. Soc., N. Ser. 17, 63–82 (1953; Zbl 0050.27102); J. Indian Math. Soc., N. Ser. 17, 83–141 (1953; Zbl 0051.28008)] and A. Selberg [J. Indian Math. Soc., N. Ser. 18, 83–87 (1954; Zbl 0057.28502)] on the distribution of primes, although Ramanujan did not state any specific theorems. In prime number theory, Dickman’s function is a famous and useful function, but in Sect. 8.2., we see that Ramanujan had discovered Dickman’s function at least 10 years before K. Dickman did in [Ark. Mat. A 22, No. 10, 14 p. (1930; JFM 56.0178.04)]. We then turn to a formula for \(\zeta(\tfrac 12)\), first given in Sect. 8 of Chap. 15 in Ramanujan’s second notebook. In (*), Ramanujan offers an elegant reinterpretation of this formula, which renders an already intriguing result even more fascinating. Next, we examine a fragment on sums of powers that was very difficult to interpret. One of the most interesting results in the chapter yields an unusual algorithm for generating solutions to Euler’s Diophantine equation \(a^3+ b^3= c^3+ d^3\).
Chapter 9 is devoted to discarded fragments of manuscripts and partial manuscripts concerning the divisor functions \(\sigma_k(n)\) and \(d(n)\), respectively, the sum of the \(k\)th powers of the divisors of \(n\), and the number of divisors of \(n\).
In the next chapter, Chap. 10, we prove all of the results on page 196 of (*). Two of the results evaluating certain Dirichlet series are especially interesting.
Chapter 11 contains some unusual old and new results on primes arranged in two rough, partial manuscripts. Ramanujan’s manuscripts contain several errors, and we conjecture that this work predates his departure for England in 1914. Harold Diamond helped us enormously in both interpreting and correcting the claims made by Ramanujan in the two partial manuscripts examined in Chap. 11.
In Chap. 12, we discuss a manuscript that was either intended to be a paper by itself or, more probably, was slated to be the concluding portion of Ramanujan’s paper [Messenger 45, 81–84 (1915; JFM 45.1250.01)]. The results in this paper hark back to Ramanujan’s early preoccupation with infinite series identities. Our account here includes a closer examination of two of Ramanujan’s series by Johann Thiel, to whom we are very grateful for his contributions.
Perhaps the most fascinating formula found in the three manuscripts on Fourier analysis in the handwriting of Watson is a transformation formula involving the Riemann \(\Xi\)-function and the logarithmic derivative of the gamma function in Chap. 13.
The second of the aforementioned manuscripts features integrals that possess transformation formulas like those satisfied by theta functions. Two of the integrals were examined by Ramanujan in two papers, where he considered the integrals to be analogues of Gauss’ sums, a view that we corroborate in Chap. 14. One of the integrals, to which page 198 of (*) is devoted, was not examined earlier by Ramanujan.
In the third manuscript, on Fourier analysis, which we discuss in Chap. 15, Ramanuian considers some problems on Mellin transforms.
The next three chapters pertain to some of Ramanujan’s earlier published papers. We then consider miscellaneous collections of results in classical analysis and elementary mathematics in the next two chapters.
Chapter 21 is devoted to some strange, partially incorrect claims of Ramanujan that likely originate from an early part of his career”.
This is the ninth volume in the Ramanujan notebooks series. Through nearly 20 years the care and attention to all manner of scholarly detail has been extraordinarily and consistently of the highest quality. Of course Ramanujan’s life was tragically short. Yet in having Berndt, and later Andrews and Berndt as editors he has surely been fortunate indeed.
The smallest fragments are lovingly handled here. The reviewer took particular delight in Chapter 6 on \(\gamma\), Euler’s constant, where a rapidly convergent series is found; also, in §7.3 we find the best possible Diophantine approximation of \(e^{2/a}\), where a \(a\neq 0\) is an integer. As this chapter remarks, Ramanujan found this 60 years before it appeared in the literature. – One stands in awe of this series of books.