1 research outputs found
A Dirichlet character analogue of Ramanujan's formula for odd zeta values
In 2001, Kanemitsu, Tanigawa, and Yoshimoto studied the following generalized
Lambert series, for and with some restriction on .
Recently, Dixit and the last author pointed out that this series has already
been present in the Lost Notebook of Ramanujan with a more general form.
Although, Ramanujan did not provide any transformation identity for it. In the
same paper, Dixit and the last author found an elegant generalization of
Ramanujan's celebrated identity for while extending the results
of Kanemitsu et al. In a subsequent work, Kanemitsu et al. explored another
extended version of the aforementioned series, namely,
where denotes a Dirichlet character modulo , and
with some restriction on the variable . In the current paper, we investigate
the above series for {\it any} and . We
obtain a Dirichlet character analogue of Dixit and the last author's identity
and there by derive a two variable generalization of Ramanujan's identity for
. Moreover, we establish a new identity for
analogous to Ramanujan's famous identity for .Comment: 24 pages, comments are welcome