Abstract
In his lost notebook, Ramanujan stated without proofs several beautiful identities for the three classical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.
| Original language | English |
|---|---|
| Pages (from-to) | 81-114 |
| Number of pages | 34 |
| Journal | Ramanujan Journal |
| Volume | 4 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2000 Jan 1 |
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All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
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Eisenstein series in Ramanujan's lost notebook. / Berndt, Bruce C.; Chan, Heng Huat; Sohn, Jaebum; Son, Seung Hwan.
In: Ramanujan Journal, Vol. 4, No. 1, 01.01.2000, p. 81-114.Research output: Contribution to journal › Article
TY - JOUR
T1 - Eisenstein series in Ramanujan's lost notebook
AU - Berndt, Bruce C.
AU - Chan, Heng Huat
AU - Sohn, Jaebum
AU - Son, Seung Hwan
PY - 2000/1/1
Y1 - 2000/1/1
N2 - In his lost notebook, Ramanujan stated without proofs several beautiful identities for the three classical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.
AB - In his lost notebook, Ramanujan stated without proofs several beautiful identities for the three classical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.
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UR - http://www.scopus.com/inward/citedby.url?scp=0034403881&partnerID=8YFLogxK
U2 - 10.1023/A:1009834123932
DO - 10.1023/A:1009834123932
M3 - Article
AN - SCOPUS:0034403881
VL - 4
SP - 81
EP - 114
JO - Ramanujan Journal
JF - Ramanujan Journal
SN - 1382-4090
IS - 1
ER -