Introduction
1. Double Series of Bessel Functions and the Circle and Divisor Problems
2. Koshliakov’s Formula and Guinand’s Formula
3. Theorems Featuring the Gamma Function
4. Hypergeometric Series
5. Two Partial Manuscripts on Euler’s Constant γ
6. Problems in Diophantine Approximation
7. Number Theory
8. Divisor Sums
9. Identities Related to the Riemann Zeta Function and Periodic Zeta Functions
10. Two Partial Unpublished Manuscripts on Sums Involving Primes
11. An Unpublished Manuscript of Ramanujan on Infinite Series Identities
12. A Partial Manuscript on Fourier and Laplace Transforms
13. Integral Analogues of Theta Functions and Gauss Sums
14. Integral Analogues of Theta Functions and Gauss Sums
15. Functional Equations for Products of Mellin Transforms
16. A Preliminary Version of Ramanujan’s Paper
17. A Preliminary Version of Ramanujan’s Paper
18. A Partial Manuscript Connected with Ramanujan’s Paper “Some Definite Integrals”
19. Miscellaneous Results in Analysis3
20. Elementary Results
21. A Strange, Enigmatic Partial Manuscript

This volume is the fourth of five volumes that the authors plan to write on Ramanujan’s lost notebook.​ In contrast to the first three books on Ramanujan's Lost Notebook, the fourth book does not focus on q-series. Most of the entries examined in this volume fall under the purviews of number theory and classical analysis. Several incomplete manuscripts of Ramanujan published by Narosa with the lost notebook are discussed. Three of the partial manuscripts are on diophantine approximation, and others are in classical Fourier analysisand prime number theory. Most of the entries in number theory fall under the umbrella of classical analytic number theory. Perhaps the most intriguing entries are connected with the classical, unsolved circle and divisor problems.