Bibekananda Maji

Indian Institute of Technology Indore | IITI · Department of Mathematics

We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for $\textup{spt}(n)$. The latter motivates us to extend the theory of the restricted partition function $p...

Zagier, in 1981, conjectured that the constant term of an automorphic function associated to the Ramanujan delta function, i.e. [Formula: see text] has a connection with the nontrivial zeros of [Formula: see text]. This conjecture was finally proved by Hafner and Stopple in 2000. Recently, Chakraborty et al. extended this observation for any normal...

In 1916, Riesz gave an equivalent criterion for the Riemann hypothesis (RH). Inspired from Riesz's criterion, Hardy and Littlewood showed that RH is equivalent to the following bound: \begin{align*} P_1(x):= \sum_{n=1}^\infty \frac{\mu(n)}{n} \exp\left({-\frac{x}{n^2}}\right) = O_{\epsilon}\left( x^{-\frac{1}{4}+ \epsilon } \right), \quad \mathrm{...

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 Eisenstein series over the full modular group. Recently, Banerjee, Gupta and Kumar found a number field analogue of Ra...

In I981, Uchimura studied a divisor generating $q$-series that has applications in probability theory and in the analysis of data structures, called heaps. Mainly, he proved the following identity. For $|q|<1$, \begin{equation*} \sum_{n=1}^\infty n q^n (q^{n+1})_\infty =\sum_{n=1}^{\infty} \frac{(-1)^{n-1} q^{\frac{n(n+1)}{2} } }{(1-q^n) ( q)_n }...

In 2000, Hafner and Stopple proved a conjecture of Zagier which states that the constant term of the automorphic function |Δ(x+iy)|2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin...

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers [Formula: see text] such that [Formula: see text] and [Formula: see text] are all sums of two squares where [Formula: see text] and [Formula: see text] are two arbitrary integers, and as an...

Utilizing inverse Mellin transform of the symmetric square $L$-function attached to Ramanujan tau function, Hafner and Stopple proved a conjecture of Zagier, which states that the constant term of the automorphic function $y^{12}|\Delta(z)|^2$ i.e., the Lambert series $y^{12}\sum_{n=1}^\infty \tau(n)^2 e^{-4 \pi n y}$ can be expressed in terms of...

Using hyperbola method, Dirichlet, in 1849, proved that the error term in the study of the summatory function of the divisor function d(n) is \(O(\sqrt{x})\). Then in 1904, Voronoï used the method of contour integration to improve the error term as \(O( x^{\frac{1}{3} + \epsilon })\), for any positive \(\epsilon \). Recently, Gupta and Maji (J. Mat...

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers n such that n, n + h and n + k are all sums of two squares where h and k are two arbitrary integers, and as an immediate corollary obtain, in parametric terms, three consecutive integers th...

In 2000, Hafner and Stopple proved a conjecture of Zagier which states that the constant term of the automorphic function |∆(x+iy)|² i.e., the Lambert series Σ∞n=1 τ (n)²e−4πny can be expressed in terms of the non-trivial zeros of the Riemann zeta function. In this article, we study an asymptotic expansion of a generalized version of the aforementi...

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, Dixit and the last author pointed out that this series has already been present in the Lost Notebook of Ramanu...

Ramanujan's notebooks contain many elegant identities and one of the celebrated identities is a formula for $\zeta(2k+1)$. In 1972, Grosswald gave an extension of the Ramanujan's formula for $\zeta(2k+1)$, which contains a polynomial of degree $2k+2$. This polynomial is now well-known as the Ramanujan polynomial$R_{2k+1}(z)$, first studied by Gun,...

In this article, we refine a result of Andrews and Newman, that is, the sum of minimal excludants over all the partitions of a number n equals the number of partitions of n into distinct parts with two colors. As a consequence, we find congruences modulo 4 and 8 for the functions appearing in this refinement. We also conjecture three further congru...

In 2000, Hafner and Stopple proved a conjecture of Zagier which states that the constant term of the automorphic function $|\Delta(x+iy)|^2$ i.e., the Lambert series $\sum_{n=1}^\infty \tau(n)^2 e^{-4 \pi n y}$ can be expressed in terms of the non-trivial zeros of the Riemann zeta function. In this article, we study a certain Lambert series associa...

In the present paper, we prove that the generalized Riemann hypothesis for the Dirichlet L-function \(L(s,\chi )\) is equivalent to the following bound: Let \(k \ge 1\) and \(\ell \) be positive real numbers. For any \(\epsilon >0\), we have $$\begin{aligned} \sum _{n=1}^{\infty } \frac{\chi (n) \mu (n)}{n^{k}} \exp \left( - \frac{ x}{n^{\ell }}\ri...

In 1984, Bressoud and Subbarao obtained an interesting weighted partition identity for a generalized divisor function, by means of combinatorial arguments. Recently, the last three named authors found an analytic proof of the aforementioned identity of Bressoud and Subbarao starting from a q-series identity of Ramanujan. In the present paper, we re...

In his lost notebook, Ramanujan noted down many elegant identities involving divisor functions and the modified K-Bessel function, and some of them are connected with the Fourier series expansion of the non-holomorphic Eisenstein series. Recently, Cohen established interesting generalizations of some of the identities of Ramanujan. In this paper, w...

For a fixed $z\in\mathbb{C}$ and a fixed $k\in\mathbb{N}$, let $\sigma_{z}^{(k)}(n)$ 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 the well-known divisor function $\sigma_z(n)$ as well as the divisor function $d^{(k)}(n)$ first studied b...

Zagier, in 1981, conjectured that the constant term of an automorphic function associated to the Ramanujan delta function, that is, $y^{12} \sum_{n=1}^{\infty}\tau^2(n) e^{-4\pi n y}$ has a connection with the non-trivial zeros of $\zeta(s)$. This conjecture was finally proved by Hafner and Stopple in 2000. Recently, Chakraborty, Kanemitsu and the...

One of the celebrated formulas of Ramanujan is about odd zeta values, which has been studied by many mathematicians over the years. A notable extension was given by Grosswald in 1972. Following Ramanujan’s idea, we rediscovered a Ramanujan-type identity for ζ(2k+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{a...

We find an interesting refinement of a result due to Andrews and Newman, that is, the sum of minimal excludants over all the partitions of a number n equals the number of partitions of n into distinct parts with two colors. As a consequence, we find congruences modulo 4 and 8 for the functions appearing in this refinement. We also conjecture three...

In 1984, Bressoud and Subbarao obtained an interesting weighted partition identity for a generalized divisor function, by means of combinatorial arguments. Recently, the last three named authors found an analytic proof of the aforementioned identity of Bressoud and Subbarao starting from a q-series identity of Ramanujan. In the present paper, we re...

In the present paper, we prove that the generalized Riemann hypothesis for the Dirichlet $L$-function $L(s,\chi)$ is equivalent to the following bound: Let $k \geq 1$ and $\ell$ be positive real numbers. For any $\epsilon >0$, we have \begin{align*} \sum_{n=1}^{\infty} \frac{\chi(n) \mu(n)}{n^{k}} \exp \left(- \frac{ x}{n^{\ell}}\right) = O_{\epsil...

Euler's classical identity states that the number of partitions of an integer into odd parts and distinct parts are equinumerous. Franklin gave a generalization by considering partitions with exactly $j$ different multiples of $r$, for a positive integer $r$. We prove an analogue of Franklin's identity by studying the number of partitions with $j$...

Hafner and Stopple proved a conjecture of Zagier that the inverse Mellin transform of the symmetric square [Formula: see text]-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function [Formula: see text]. Later, Chakraborty et al. extended this phenomenon for any Hec...

Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five identities do not seem to have acquired enough attention. Recently, Dixit and the third author found a one-variable generalization of one of the aforementioned five identities. From the...

Recently, Juyal, Maji, and Sathyanarayana have studied a Lambert series associated with a cusp form over the full modular group and the Möbius function. In this paper, we investigate the Lambert series ∑n=1∞[af(n)ψ(n)∗μ(n)ψ′(n)]exp(-ny),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb...

The average size of the “smallest gap” of a partition was studied by Grabner and Knopfmacher in 2006. Recently, Andrews and Newman, motivated by the work of Fraenkel and Peled, studied the concept of the “smallest gap” under the name “minimal excludant” of a partition and rediscovered a result of Grabner and Knopfmacher. In this paper, we study the...

Page 332 of Ramanujan's Lost Notebook contains a compelling identity for ζ(1/2), which has been studied by many mathematicians over the years. On the same page, Ramanujan also recorded the series,1rexp⁡(1sx)−1+2rexp⁡(2sx)−1+3rexp⁡(3sx)−1+⋯, where s is a positive integer and r−s is any even integer. Unfortunately, Ramanujan doesn't give any formula...

In 1981, Zagier conjectured that the constant term of the automorphic form \( y^{12}|\Delta (z)|^2\), that is, \( a_0(y):=y^{12} \sum _{n=1}^{\infty } \tau ^2(n) \exp ({-4 \pi ny}), \) where \(\tau (n)\) is the nth Fourier coefficient of the Ramanujan cusp form \(\Delta (z)\), has an asymptotic expansion when \(y \rightarrow 0^{+}\) and it can be e...

In 1916, Riesz proved that the Riemann hypothesis is equivalent to the bound $ \sum_{n=1}^\infty \frac{\mu(n)}{n^2} \exp\left( - \frac{x}{n^2} \right) = O_{\epsilon} \left( x^{-\frac{3}{4} + \epsilon} \right), $ as $x \rightarrow\infty$, for any $\epsilon >0$. Around the same time, Hardy and Littlewood gave another equivalent criteria for the Rie...

One of the celebrated formulas of Ramanujan is about odd zeta values, which has been studied by many mathematicians over the years. A notable extension was given by Grosswald in 1972. Following Ramanujan's idea, we rediscovered a Ramanujan-type identity for $\zeta(2k+1)$ that was first established by Malurkar and later by Berndt using different tec...

We find an interesting refinement of a result due to Andrews and Newman, that is, the sum of minimal excludants over all the partitions of a number n equals the number of distinct parts partitions of $n$ into two colors. In addition, we also study $k$th moments of minimal excludants by means of generalizing the aforementioned result. At the end, we...

Recently, Juyal, Maji, and Sathyanarayana have studied a Lambert series associated with a cusp form over the full modular group and the Möbius function. In this paper, we investigate the Lambert series $\sum_{n=1}^{\infty}[a_f(n)\psi(n)*\mu(n)\psi'(n)]\exp(-ny)$, where $a_f (n)$ is the nth Fourier coefficient of a cusp form f over any congruence su...

Page 332 of Ramanujan's Lost Notebook contains a compelling identity for ζ(1/2), which has been studied by many mathematicians over the years. On the same page, Ramanujan also recorded the series, \begin{equation*} \frac{1^r}{\exp(1^s x) - 1} + \frac{2^r}{\exp(2^s x) - 1} + \frac{3^r}{\exp(3^s x) - 1} + \cdots, \end{equation*} where s is a positiv...

The average size of the "smallest gap" of a partition was studied by Grabner and Knopfmacher in 2006. Recently, Andrews and Newman, motivated by the work of Fraenkel and Peled, studied the concept of the "smallest gap" under the name "minimal excludant" of a partition and rediscovered a result of Grabner and Knopfmacher. In the present paper, we st...

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 zeta function ζ(s). Later, Chakraborty, Kanemitsu and the second author extended this phenomenon for any Hecke ei...

Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five identities do not seem to have acquired enough attention. Recently, Dixit and the third author found a one-variable generalization of one of the aforementioned five identities. From the...

In 1981, Zagier conjectured that the constant term of the automorphic form $ y^{12}|\Delta(z)|^2$, that is, $ a_0(y):=y^{12} \sum_{n=1}^{\infty} \tau^2(n) \exp({-4 \pi ny}),$ where $\tau(n)$ is the $n$th Fourier coefficient of the Ramanujan cusp form $\Delta(z)$, has an asymptotic expansion when $y \rightarrow 0^{+}$ and it can be expressed in te...

A generalization of a beautiful q-series identity found in the unorganized portion of Ramanujan's second and third notebooks is obtained. As a consequence, we derive a new three-parameter identity which is a rich source of partition-theoretic information. In particular, we use this identity to obtain a generalization of a recent result of Andrews,...

Recently, Lalín, Rodrigue, and Rogers have studied the secant zeta function and its convergence. They found many interesting values of the secant zeta function at some particular quadratic irrational numbers. They also gave modular transformation properties of the secant zeta function. In this paper, we generalized secant zeta function as a Lambert...

In this paper, we consider the problem about finding out perfect powers in an alternating sum of consecutive cubes. More precisely, we completely solve the Diophantine equation $(x+1)^3 - (x+2)^3 + \cdots - (x + 2d)^3 + (x + 2d + 1)^3 = z^p$, where $p$ is prime and $x,d,z$ are integers with $1 \leq d \leq 50$.

The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite analogues of rank and crank moments for vector partitions while deriving a finite analogue of Andrews' famous identity for smallest parts function. In the same paper, they also conjectured an in...

It is pointed out that the generalized Lambert series $\sum\nolimits_{n = 1}^\infty {[(n^{N-2h})/(e^{n^Nx}-1)]} $ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by re...

We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for $\textup{spt}(n)$. The latter motivates us to extend the theory of the restricted partition function $p...

A comprehensive study of the generalized Lambert series $\sum _{n=1}^{\infty }\frac{n^{N-2h}\text{exp}(-an^{N}x)}{1-\text{exp}(-n^{N}x)},00$ , $N\in \mathbb{N}$ and $h\in \mathbb{Z}$ , is undertaken. Several new transformations of this series are derived using a deep result on Raabe’s cosine transform that we obtain here. Three of these transformat...

A generalization of a beautiful $q$-series identity found in the unorganized portion of Ramanujan's second and third notebooks is obtained. As a consequence, we derive a new three-parameter identity which is a rich source of partition-theoretic information. In particular, we use this identity to obtain a generalization of a recent result of Andrews...

A comprehensive study of the generalized Lambert series ∞ n=1 n N −2h exp(−an N x) 1 − exp(−n N x) , 0 < a ≤ 1, x > 0, N ∈ N and h ∈ Z, is undertaken. Two of the general transformations of this series that we obtain here lead to two-parameter generalizations of Ramanujan's famous formula for ζ(2m + 1), m > 0 and the transformation formula for log η...

In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral representation involving the Riemann $\Xi$-function and the confluent hypergeometric function linked to the general theta tra...

In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral representation involving the Riemann $\Xi$-function and the confluent hypergeometric function linked to the general theta tra...

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 general form. Kanemitsu, Tanigawa and Yoshimoto obtained an important transformation of this series for the para...

In this paper, we consider the problem about finding out perfect powers in an alternating sum of consecutive cubes. More precisely, we completely solve the Diophantine equation $(x+1)^3 - (x+2)^3 + \cdots - (x + 2d)^3 + (x + 2d + 1)^3 = z^p$, where $p$ is prime and $x,d,z$ are integers with $1 \leq d \leq 50$.

Zagier's conjecture on the asymptotic expansion of the Lambert series $ \sum_{n=1}^{\infty} \tau^2(n)\exp(-nz),$ where $\tau(n)$ is the Ramanujan's tau function, was proved by Hafner and Stopple. Recently, Chakraborty, Kanemitsu and Maji have extended this result to any cusp forms over the full modular group. The goal of this paper is to extend th...

Hafner and Stopple proved a conjecture of Zagier relating to the asymptotic behavior of the inverse Mellin transform of the symmetric square L-function associated to Ramanujan tau function. In this paper we prove similar result for any cusp form over the full modular group.

The Abel–Tauber process consists of the Abelian process of forming the Riesz sums and the subsequent Tauberian process of differencing the Riesz sums, an analogue of the integration–differentiation process. In this article, we use the Abel–Tauber process to establish an interesting asymptotic expansion for the Riesz sums of arithmetic functions wit...

Many authors have studied the problem of finding sequences of rational points on elliptic curves such that either the abscissae or the ordinates of these points are in arithmetic progression. In this paper we obtain upper bounds for the lengths of sequences of rational points on curves of the type $y^2=x^3+k$, $k \in \mathbb{Q} \setminus \{0\}$, su...

Abel-Tauber process consists of the Abelian process of forming the Riesz sums and the subsequent Tauberian process of differencing the Riesz sums, an analogue of the integration--differentiation process. In this article, we use Abel-Tauber process to establish interesting asymptotic expansion for the Riesz sums of arithmetic functions with best pos...

In this article, we will discuss some interesting proofs of the infinitude of prime numbers and provide a new way to construct an infinite sequence of pairwise relatively prime natural numbers.