arXiv subjects

Diksha Rani Bansal

Publications and source records attributed to Diksha Rani Bansal.

5 recordsLinked to original sources

arXiv2026

A number field extension of Ramanujan's identity and its integral representation via the generalized Riemann $Ξ$-function

On pages 219--220 of Ramanujan's Lost Notebook appears an intriguing modular transformation whose integral representation is expressed in terms of the Riemann $Ξ$-function. In this paper, we establish a number field analogue of this identity. We also obtain an integral representation for the transformation formula associated with the functional equation of the higher power of the Dedekind zeta function recently studied by the present author and Maji. Interestingly, we recover Hardy's integral representation of the well-known Jacobi theta relation involving the Riemann $Ξ$-function. We further obtain a number-field analogue of the extended Ramanujan--Koshliakov formula established by Dixit.

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arXiv2026

An extension of Ramanujan-Guinand identity for the Dedekind zeta function and a new formula for $ζ_\mathbb{F}(2)$ and $ζ_\mathbb{F}(2m+1)$

On page 253 of his Lost Notebook, Ramanujan recorded an intriguing identity relating a generalized divisor function and a modified Bessel function, which was later rediscovered by Guinand and is now known as the Ramanujan-Guinand identity. In this paper, we establish a number field analogue of this identity for the Dedekind zeta function. In particular, we recover Ramanujan-Guinand identity, Ramanujan-Koshliakov identity, and also obtain the transformation formula for the logarithm of the Dedekind eta function. In 1986, Zagier obtained a formula for $ζ_\mathbb{F}(2)$ for any number field $\mathbb{F}$. Quite surprisingly, as an application of our main theorem, we also obtain a new formula for $ζ_\mathbb{F}(2)$ for arbitrary number field $\mathbb{F}$. Moreover, we derive an elegant identity for $ζ_\mathbb{F}(2m+1)$ for any positive integer $m$ and any number field $\mathbb{F}$.

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arXiv2025

Analogues of Harglotz-Zagier-Novikov function

Recently, Choie and Kumar extensively studied the Herglotz-Zagier-Novikov function $\mathfrak{F}(z;u,v)$, defined as \begin{align*} \mathfrak{F}(z;u,v) = \int_{0}^{1} \frac{\log(1-ut^z)}{v^{-1}-t} dt, \quad \textrm{for} \quad \mathfrak{Re}(z)>0. \end{align*} They obtained two-term, three-term and six-term functional equations for $\mathfrak{F}(z;u,v)$ and also evaluated special values in terms of di-logarithmic functions. Motivated from their work, we study the following two integrals, \begin{align*} \mathfrak{F}(z;u,v,w) &=\int_{0}^1 \frac{\log(1-ut^z)\log(1-wt^z)}{v^{-1}-t}\text{d}t, \\ \mathfrak{F}_k(z;u,v) &= \int_{0}^{1} \frac{\log^k(1-ut^z)}{v^{-1}-t} \, \text{d}t, \end{align*} for $\mathfrak{Re}(z)>0$ and $k \in \mathbb{N}$. For $k=1$, the integral $\mathfrak{F}_k(z;u,v)$ reduces to $\mathfrak{F}(z;u,v)$. This allows us to recover the properties of $\mathfrak{F}(z;u,v)$ by studying the properties of $\mathfrak{F}_k(z;u,v)$. We evaluate special values of these two functions in terms of poly-logarithmic functions.

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arXiv2025

Number Field Analogue of Jacobi Theta Relation And Zeros of Dedekind zeta function on Re$(s)=1/2$

In 1914, Hardy proved that there are infinitely many non-trivial zeros of the Riemann zeta function $ζ(s)$ on the critical line Re$(s)=1/2$ using the Jacobi theta relation. In this paper, we first establish a number field analogue of the Jacobi theta relation and as an application, we show the existence of infinitely many non-trivial zeros of the Dedekind zeta function $ζ_\mathbb{F}(s)$ on Re$(s)=1/2$, for any number field $\mathbb{F}$. Quite interestingly, we also prove that the Jacobi theta relation is equivalent to an intriguing identity of Hardy, Littlewood and Ramanujan.

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arXiv2024

A Number Field Analogue of Ramanujan's identity for $ζ(2m+1)$

Ramanujan's famous formula for $ζ(2m+1)$ 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 Eisenstein series over the full modular group. Recently, Banerjee, Gupta and Kumar found a number field analogue of Ramanujan's formula. In this paper, we present a new number field analogue of the Ramanujan-Grosswald formula for $ζ(2m+1)$ by obtaining a formula for Dedekind zeta function at odd arguments. We also obtain a number field analogue of an identity of Chandrasekharan and Narasimhan, which played a crucial role in proving our main identity. As an application, we generalize transformation formula for Eisenstein series $G_{2k}(z)$ and Dedekind eta function $η(z)$. A new formula for the class number of a totally real number field is also obtained, which provides a connection with the Kronceker's limit formula for the Dedekind zeta function.

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