Related papers: Generalized Lambert series, Raabe's integral and a…
It is pointed out that the generalized Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}}{e^{n^{N}x}-1}$ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page $332$ of Ramanujan's Lost Notebook in a slightly more…
A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for $\zeta(2m+1)$. The…
In 2001, Kanemitsu, Tanigawa, and Yoshimoto studied the following generalized Lambert series, $$ \sum_{n=1}^{\infty} \frac{n^{N-2h} }{\exp(n^N x)-1}, $$ for $N \in \mathbb{N}$ and $h\in \mathbb{Z}$ with some restriction on $h$. Recently,…
Let $\sigma_a^{(N)}(n)=\sum_{d^{N}|n}d^a$. An explicit transformation is obtained for the generalized Lambert series $\sum_{n=1}^{\infty}\sigma_{a}^{(N)}(n)e^{-ny}$ for Re$(a)>-1$ using the recently established Vorono\"i summation formula…
We shall make use of the method of partial fractions to generalize some of Ramanujan's infinite series identities, including Ramanujan's famous formula for $\zeta(2n+1)$, and we shall also give a generalization of the transformation formula…
Page 332 of Ramanujan's Lost Notebook contains a compelling identity for $\zeta(1/2)$, which has been studied by many mathematicians over the years. On the same page, Ramanujan also recorded the series, \begin{align*} \frac{1^r}{\exp(1^s x)…
An exact transformation, which we call the \emph{master identity}, is obtained for the first time for the series $\sum_{n=1}^{\infty}\sigma_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. New modular-type transformations when $a$ is a…
In this article, we have studied transformation formulas of zeta function at odd integers over an arbitrary number field which in turn generalizes Ramanujan's identity for the Riemann zeta function. The above transformation leads to a new…
Ramanujan's famous formula for $\zeta(2m+1)$ has captivated the attention of numerous mathematicians over the years. Grosswald, in 1972, found a simple extension of Ramanujan's formula which in turn gives transformation formula for…
Ramanujan made many beautiful and elegant discoveries in his short life of 32 years, and one of them that has attracted the attention of several mathematicians over the years is his intriguing formula for $\zeta(2n+1)$. To be sure,…
We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x^2+5y^2. Making use of Ramanujan's_1\psi_1 summation formula we establish a new Lambert series identity for…
In his second notebook, Ramanujan discovered the following identity for the special values of $\zeta(s)$ at the odd positive integers \begin{equation*}\begin{aligned}\alpha^{-m}\,\left\{\dfrac{1}{2}\,\zeta(2m + 1) + \sum_{n =…
In this paper, we find rational zeta series with $\zeta(2n)$ in terms of $\zeta(2k+1)$ and $\beta(2k)$, the Dirichlet beta function. We then develop a certain family of generalized rational zeta series using the generalized Clausen function…
In this article, we obtain a transformation formula for the higher power of odd zeta values, which generalizes Ramanujan's formula for odd zeta values. We have also investigated many important applications, which in turn provide…
We consider integral and series transformations, which are associated with Ramanujan's identities, involving various arithmetic functions and a ratio of products of Riemann's zeta functions of different arguments. Reciprocal inversion…
An explicit transformation for the series $\sum\limits_{n=1}^{\infty}\displaystyle\frac{\log(n)}{e^{ny}-1},$ Re$(y)>0$, which takes $y$ to $1/y$, is obtained for the first time. This series transforms into a series containing $\psi_1(z)$,…
The author derives new family of series representations for the values of the Riemann Zeta function $\zeta(s)$ at positive odd integers. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with…
The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application…
Ramanujan derived the well known divergent-sum of integers in more than one way. We generalise the informal method to higher powers of the Riemann zeta function through a study of the Eulerian numbers in particular. Within the context of…
Hafner and Stopple proved a conjecture of Zagier, that the inverse Mellin transform of the symmetric square $L$-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the non-trivial zeros of the Riemann…