arXiv subjects

Debika Banerjee

Publications and source records attributed to Debika Banerjee.

8 recordsLinked to original sources

arXiv2026

On $\mathcal{B}^4$-almost periodicity for a class of arithmetical functions

In this paper, we establish the $\mathcal{B}^4$-almost periodicity in the sense of Besicovitch for a suitably normalized error term associated with a broad class of arithmetical functions introduced by Chandrasekharan and Narasimhan. This result significantly strengthens $\mathcal{B}^2$-almost periodicity previously investigated in the literature for related error terms. By deriving truncated Vorono\"{i}-type formulas, we demonstrate that the normalized error term lies in the Besicovitch space $B^4$. As a consequence, we deduce that the error term admits a limit probability distribution and establish an explicit formula for its mean fourth power moment.

math.NT↗

arXiv2026

A divisor function of Wigert and higher degree forms

Let $k\in\mathbb{N}$. Wigert's divisor function $d^{\left(\frac{1}{k}\right)}(j)$ counts the number of representations of $j$ of the form $m^k+mn$ with $m\geq1 , n\geq0$. Let $\mathcal{F}_k(s)$ denote the Dirichlet series of $d^{\left(\frac{1}{k}\right)}(j)$. While $\mathcal{F}_2(s)$ is essentially a well-known special case of the Euler-Zagier double zeta function, and hence well-studied, very little is known about $\mathcal{F}_k(s)$ for $k>2$. We offer three new representations for $\mathcal{F}_k(s)$ for $k\geq2$, one of which is an analogue of the Chowla-Selberg formula as well as of a formula of Atkinson. The meromorphicity of $\mathcal{F}_k(s)$ is also discussed. The special value $\mathcal{F}_3\left(\frac{3}{2}\right)$ is expressed in terms of an infinite series of Bessel functions and a generalized divisor function.

math.NT↗

arXiv2025

Divisibility properties of weighted $k$ regular partitions

We study a generalized class of weighted $k$-regular partitions defined by \[ \sum_{n=0}^{\infty} c_{k, r_1, r_2}(n) q^n = \prod_{n=1}^{\infty} \frac{(1 - q^{nk})^{r_1}}{(1 - q^n)^{r_2}}, \] which extends the classical $k$-regular partition function $b_k(n)$. We establish new infinite families of Ramanujan-type congruences, divisibility results, and positive-density prime sets for which $c_{k, r_1, r_2}(n)$ vanishes modulo a given prime.

math.NT↗

arXiv2025

A weighted divisor problem and exponential sum

In this paper, we investigate a weighted divisor problem involving the exponential sum of $D_{(1)}(n)$, the $n$th coefficient in the Dirichlet series expansion of $\zeta'(s)^2$. We establish a truncated Vorono\"{i} type formula for the error term of $\sum_{n\leq x}D_{(1)}(n)e(nh/k)$, analogous to the results obtained by Jutila. Utilizing this truncated formula, we derive a mean square estimate of the error term. In addition, we study the associated Riesz sum and the corresponding error term, along with its mean square estimate.

math.NT↗

arXiv2023

Trigonometric analogue of the identities associated with twisted sums of divisor functions

Inspired by two entries published in Ramanujan's lost notebook on Page 355, B. C. Berndt et al.\cite{MR3351542} presented Riesz sum identities for Ramanujan entries by introducing the twisted divisor sums. Later, S. Kim \cite{MR3541702} derived analogous results by replacing twisted divisor sums with twisted sums of divisor functions. Recently, the authors \cite{devika2023} of the present paper deduced the Cohen-type identities as well as Vorono\"i summation formulas associated with these twisted sums of divisor functions. The present paper aims to derive an equivalent version of the results in the previous paper in terms of identities involving finite sums of trigonometric functions and the doubly infinite series. As an application, the authors provide an identity for $r_6(n)$, which is analogous to Hardy's famous result where $r_6(n)$ denotes the number of representations of natural number $n$ as a sum of six squares.

math.NT↗

arXiv2023

Character analogues of Cohen type identities and related Voronoi summation formulas

In \cite{MR2221114}, B.~C.~Berndt and A.~Zaharescu introduced the twisted divisor sums associated with the Dirichlet character while studying the Ramanujan's type identity involving finite trigonometric sums and doubly infinite series of Bessel functions. Later, in a follow-up paper \cite{MR3541702}, S. Kim extended the definition of the twisted divisor sums to twisted sums of divisor functions. In this paper, we derive identities associated with the aforementioned weighted divisor functions and the modified $K$-Bessel function in light of recent results obtained by the first author and B. Maji \cite{ debika2023}. Moreover, we provide a new expression for $L(1, \chi)$ from which we establish the positivity of $L(1, \chi)$ for any real primitive character $\chi$. In addition, we deduce Cohen-type identities and then exhibit the Vorono\"i-type summation formulas for them.

math.NT↗

arXiv2023

Distribution of values of general Euler totient function

Let $\Phi_k(n)=|\{ (x_1, x_2, \cdots, x_k)\in \left(\mathbb{Z}/n\mathbb{Z}\right)^k; \ \gcd(x_1^2+x_2^2+ \cdots+ x_k^2, n)=1\}|$ be a general totient function introduced first by Cald\'{e}ron et. al. Motivated by the classical works of Schoenberg, Erd\H{o}s, Bateman and Diamond on the distribution of $\Phi_1(n)$, we prove results on the joint distribution of $\Phi_k(n)$ for any $k\ge 1$. Additionally, we also exhibit the extremal order of $\Phi_k(n)$.

math.NT↗

arXiv2017

Study of the rare decays $B_(s,d)^*$ arrow $l^+ l^-$ in Z' model

The rare decays $B_{(s,d)}^*\to l^+ l^-$ are important to probe the flavour sector of the standard model and to search new physics beyond the SM. Unlike pseudoscalar B meson, the leptonic decays of vector $ B_(s,d)^*$ mesons are not chirally supressed which compensates for their short lifetimes, and results significant branching ratios. In this paper, we estimate the branching ratios of $B_{(s,d)}^*\to l^+ l^-$ $(l=e,μ)$ rare decays in Z' model which is an extension of the SM with an extra U(1)' gauge symmetry. We find that the branching ratios are increased from their corresponding standard model values and vary with the mass of $Z'$ boson. Lower is the mass of $Z'$ boson, higher is the branching ratio.

hep-ph↗