Atul Dixit
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View article: On a function of Ramanujan twisted by a logarithm
On a function of Ramanujan twisted by a logarithm Open
A two-term functional equation for an infinite series involving the digamma function and a logarithmic factor is derived. A modular relation on page 220 of Ramanujan's Lost Notebook as well as a corresponding recent result for the derivati…
View article: The Rogers-Ramanujan dissection of a theta function
The Rogers-Ramanujan dissection of a theta function Open
Page 27 of Ramanujan's Lost Notebook contains a beautiful identity which not only gives, as a special case, a famous modular relation between the Rogers-Ramanujan functions $G(q)$ and $H(q)$ but also a relation between two fifth order mock…
View article: Voronoi summation formulas, oscillations of Riesz sums, and Ramanujan-Guinand and Cohen type identities
Voronoi summation formulas, oscillations of Riesz sums, and Ramanujan-Guinand and Cohen type identities Open
We derive Vorono\"{\dotlessi} summation formulas for the Liouville function $λ(n)$, the Möbius function $μ(n)$, and for $d^{2}(n)$, where $d(n)$ is the divisor function. The formula for $λ(n)$ requires explicit evaluation of certain infini…
View article: Mordell-Tornheim zeta functions and functional equations for Herglotz-Zagier type functions
Mordell-Tornheim zeta functions and functional equations for Herglotz-Zagier type functions Open
The Mordell-Tornheim zeta function and the Herglotz-Zagier function $F(x)$ are two important functions in Mathematics. By generalizing a special case of the former, namely $Θ(z, x)$, we show that the theories of these functions are inextri…
View article: Recent developments pertaining to Ramanujan's formula for odd zeta values
Recent developments pertaining to Ramanujan's formula for odd zeta values Open
In this expository article, we discuss the contributions made by several mathematicians with regard to a famous formula of Ramanujan for odd zeta values. The goal is to complement the excellent survey by Berndt and Straub \cite{berndtstrau…
View article: Modular relations involving generalized digamma functions
Modular relations involving generalized digamma functions Open
Generalized digamma functions $ψ_k(x)$, studied by Ramanujan, Deninger, Dilcher, Kanemitsu, Ishibashi etc., appear as the Laurent series coefficients of the zeta function associated to an indefinite quadratic form. In this paper, a modular…
View article: Explicit transformations for generalized Lambert series associated with the divisor function $σ_{a}^{(N)}(n)$ and their applications
Explicit transformations for generalized Lambert series associated with the divisor function $σ_{a}^{(N)}(n)$ and their applications Open
Let $σ_a^{(N)}(n)=\sum_{d^{N}|n}d^a$. An explicit transformation is obtained for the generalized Lambert series $\sum_{n=1}^{\infty}σ_{a}^{(N)}(n)e^{-ny}$ for Re$(a)>-1$ using the recently established Voronoï summation formula for $σ_a^{(N…
View article: Voronoi summation formula for the generalized divisor function $σ_{z}^{(k)}(n)$
Voronoi summation formula for the generalized divisor function $σ_{z}^{(k)}(n)$ Open
For a fixed $z\in\mathbb{C}$ and a fixed $k\in\mathbb{N}$, let $σ_{z}^{(k)}(n)$ denote the sum of $z$-th powers of those divisors $d$ of $n$ whose $k$-th powers also divide $n$. This arithmetic function is a simultaneous generalization of …
View article: Modified Bessel Functions in Analytic Number Theory
Modified Bessel Functions in Analytic Number Theory Open
The modified Bessel functions $K_ν(z)$, or, for brevity, K-Bessel functions, arise at key places in analytic number theory. In particular, they appear in beautiful arithmetic identities. A survey of these arithmetical identities and their …
View article: A modular relation involving a generalized digamma function and asymptotics of some integrals containing Ξ(t)
A modular relation involving a generalized digamma function and asymptotics of some integrals containing Ξ(t) Open
A modular relation of the form $F(\alpha, w)=F(\beta, iw)$, where $i=\sqrt{-1}$ and $\alpha\beta=1$, is obtained. It involves the generalized digamma function $\psi_w(a)$ which was recently studied by the authors in their work on developin…
View article: A modular relation involving a generalized digamma function and asymptotics of some integrals containing $Ξ(t)$
A modular relation involving a generalized digamma function and asymptotics of some integrals containing $Ξ(t)$ Open
A modular relation of the form $F(α, w)=F(β, iw)$, where $i=\sqrt{-1}$ and $αβ=1$, is obtained. It involves the generalized digamma function $ψ_w(a)$ which was recently studied by the authors in their work on developing the theory of the g…
View article: Applications of Lipschitz summation formula and a generalization of Raabe's cosine transform
Applications of Lipschitz summation formula and a generalization of Raabe's cosine transform Open
General summation formulas have been proved to be very useful in analysis, number theory and other branches of mathematics. The Lipschitz summation formula is one of them. In this paper, we give its application by providing a new transform…
View article: A finite analogue of a $q$-series identity of Bhoria, Eyyunni and Maji and its applications
A finite analogue of a $q$-series identity of Bhoria, Eyyunni and Maji and its applications Open
Bhoria, Eyyunni and Maji recently obtained a four-parameter $q$-series identity which gives as special cases not only all five entries of Ramanujan on pages 354 and 355 of his second notebook but also allows them to obtain an analytical pr…
View article: A modular relation involving non-trivial zeros of the Dedekind zeta function, and the Generalized Riemann Hypothesis
A modular relation involving non-trivial zeros of the Dedekind zeta function, and the Generalized Riemann Hypothesis Open
We give a number field analogue of a result of Ramanujan, Hardy and Littlewood, thereby obtaining a modular relation involving the non-trivial zeros of the Dedekind zeta function. We also provide a Riesz-type criterion for the Generalized …
View article: Lambert series of logarithm, the derivative of Deninger's function $R(z)$ and a mean value theorem for $ζ\left(\frac{1}{2}-it\right)ζ'\left(\frac{1}{2}+it\right)$
Lambert series of logarithm, the derivative of Deninger's function $R(z)$ and a mean value theorem for $ζ\left(\frac{1}{2}-it\right)ζ'\left(\frac{1}{2}+it\right)$ Open
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 $ψ_1(z)$, the…
View article: Two General Series Identities Involving Modified Bessel Functions and a Class of Arithmetical Functions
Two General Series Identities Involving Modified Bessel Functions and a Class of Arithmetical Functions Open
We consider two sequences $a(n)$ and $b(n)$, $1\leq n
View article: Combinatorial identities associated with a bivariate generating function for overpartition pairs
Combinatorial identities associated with a bivariate generating function for overpartition pairs Open
We obtain a three-parameter $q$-series identity that generalizes two results of Chan and Mao. By specializing our identity, we derive new results of combinatorial significance in connection with $N(r, s, m, n)$, a function counting certain…
View article: Ramanujan and Koshliakov Meet Abel and Plana
Ramanujan and Koshliakov Meet Abel and Plana Open
The neglected Russian mathematician, N.~S.~Koshliakov, derived beautiful generalizations of the classical Abel--Plana summation formula through a setting arising from a boundary value problem in heat conduction. When we let the parameter $…
View article: A Class of Identities Associated with Dirichlet Series Satisfying Hecke's Functional Equation
A Class of Identities Associated with Dirichlet Series Satisfying Hecke's Functional Equation Open
We consider two sequences $a(n)$ and $b(n)$, $1\leq n
View article: Extended higher Herglotz functions I. Functional equations
Extended higher Herglotz functions I. Functional equations Open
In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function $F(x)$ which is now known as the \emph{Herglotz function}. As demonstrated by Zagier, and very rece…
View article: Ramanujan's Beautiful Integrals
Ramanujan's Beautiful Integrals Open
Throughout his entire mathematical life, Ramanujan loved to evaluate definite integrals. One can find them in his problems submitted to the Journal of the Indian Mathematical Society, notebooks, Quarterly Reports to the University of Madra…
View article: Generalizations of the Andrews-Yee identities associated with the mock theta functions $ω(q)$ and $ν(q)$
Generalizations of the Andrews-Yee identities associated with the mock theta functions $ω(q)$ and $ν(q)$ Open
George Andrews and Ae Ja Yee recently established beautiful results involving bivariate generalizations of the third order mock theta functions $ω(q)$ and $ν(q)$, thereby extending their earlier results with the second author. Generalizing…
View article: Explicit transformations of certain Lambert series
Explicit transformations of certain Lambert series Open
An exact transformation, which we call the \emph{master identity}, is obtained for the first time for the series $\sum_{n=1}^{\infty}σ_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. New modular-type transformations when $a$ is a non-ze…
View article: A generalized modified Bessel function and explicit transformations of certain Lambert series
A generalized modified Bessel function and explicit transformations of certain Lambert series Open
An exact transformation, which we call a \emph{master identity}, is obtained for the series $\sum_{n=1}^{\infty}\sigma_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. As corollaries when $a$ is an odd integer, we derive the well-known t…
View article: On Hurwitz zeta function and Lommel functions
On Hurwitz zeta function and Lommel functions Open
We obtain a new proof of Hurwitz's formula for the Hurwitz zeta function $ζ(s, a)$ beginning with Hermite's formula. The aim is to reveal a nice connection between $ζ(s, a)$ and a special case of the Lommel function $S_{μ, ν}(z)$. This con…