On some claims in Ramanujan’s ‘unpublished’ manuscript on the partition and tau functions
نویسنده
چکیده
We disprove various claims made by Ramanujan in his, until very recently, unpublished manuscript [1] on the partition and tau functions. Furthermore, the second part of a related paper by G.K. Stanley [17] is corrected (the first part of which was earlier corrected by D. Shanks [16]).
منابع مشابه
Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary
When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s tau-function τ(n). The first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising material for insertion in the foregoing part of the manuscript. G. H. Hardy extracted a portion of P...
متن کاملRamanujan’s Elementary Method in Partition Congruences
Page 182 in Ramanujan’s lost notebook corresponds to page 5 of an otherwise lost manuscript of Ramanujan closely related to his paper providing elementary proofs of his partition congruences p(5n + 4) ≡ 0 (mod 5) and p(7n+ 5) ≡ 0 (mod 7). The claims on page 182 are proved and discussed, and further results depending on Ramanujan’s ideas are established.
متن کاملEuler Products in Ramanujan’s Lost Notebook
In his famous paper, “On certain arithmetical functions”, Ramanujan offers for the first time the Euler product of the Dirichlet series in which the coefficients are given by Ramanujan’s tau-function. In his lost notebook, Ramanujan records further Euler products for L-series attached to modular forms, and, typically, does not record proofs for these claims. In this semi-expository article, for...
متن کاملUsing it Bundles in Published and Unpublished Writings
Lexical bundles are known as important elements of coherent discourse that have been the subject of much research. While the previous research has been mainly concerned with exploring variations in the use of these word sequences across different registers and disciplines, very few studies have addressed the use of some particular groups of lexical bundles within some types of academic writing....
متن کاملA problem in diophantine approximation found in Ramanujan ’ s lost notebook Bruce C
Pages 262–265 in Ramanujan’s Lost Notebook and Other Unpublished Papers are devoted to a draft of a manuscript on an unusual problem in diophantine approximation. Ramanujan finds the maximum value of a polynomial when the variable takes rational values, and he observes the maxima’s periodicity. Because definitions and theorems in the manuscript are imprecise, the authors have attempted to inter...
متن کامل