TY - JOUR
T1 - Euler products in ramanujan's lost notebook
AU - Berndt, Bruce C.
AU - Kim, Byungchan
AU - Williams, Kenneth S.
N1 - The first author’s research was partially supported by NSA grant MDA904-00-1-0015. The second author’s research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF 2012-0003238).
PY - 2013/8
Y1 - 2013/8
N2 - 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 the Euler products appearing in his lost notebook, we provide or sketch proofs using elementary methods, binary quadratic forms, and modular forms.
AB - 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 the Euler products appearing in his lost notebook, we provide or sketch proofs using elementary methods, binary quadratic forms, and modular forms.
KW - Dedekind eta-function
KW - Dirichlet series with Euler products
KW - Eisenstein series
KW - Ramanujan tau-function
KW - Ramanujan's lost notebook
KW - binary quadratic forms
UR - https://www.scopus.com/pages/publications/84879315631
UR - https://www.scopus.com/pages/publications/84879315631#tab=citedBy
U2 - 10.1142/S1793042113500292
DO - 10.1142/S1793042113500292
M3 - Article
AN - SCOPUS:84879315631
SN - 1793-0421
VL - 9
SP - 1313
EP - 1349
JO - International Journal of Number Theory
JF - International Journal of Number Theory
IS - 5
ER -