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Soumyarup Banerjee

Publications and source records attributed to Soumyarup Banerjee.

18 recordsLinked to original sources
arXiv2026

Laurent coefficients of Zagier-type zeta function {à} la Ishibashi and arithmetic aspects of extended Ramanujan period function

One of the remarkable contributions of Don Zagier was the Kronecker limit formula for a real quadratic field, where he connects the double series $\mathcal{Z}(s,w,w^\prime)$ to the Dedekind zeta function associated to a real quadratic field. Later, Ishibashi determined all the Laurent coefficients of $\mathcal{Z}(s,w,w^\prime)$ at $s=1$. Recently, Choie and kumar have studied the analytic behaviour of the analogous double series $\tilde{\mathcal{Z}}(s,w,w^\prime)$. In this article, we derive all the Laurent coefficients of $\tilde{\mathcal{Z}}(s,w,w^\prime)$, akin to Ishibashi. These Laurent coefficients involve an interesting function $\mathfrak{F}_k^0(x)$, which was earlier studied by Dixit et. al. (Ramanujan for $k=1$), where they obtained a beautiful symmetric relation for $\mathfrak{F}_k^0(x)$. We establish both the two term and the three term functional equation of $\mathfrak{F}_k^0(x)$, derive the action of the period-like Hecke operator on $\mathfrak{F}_k^0(x)$ and connect an important integral with $\mathfrak{F}_k^0(x)$.

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arXiv2026

Hecke-type action on higher order Herglotz-Zagier function

In a seminal paper, Lewis and Zagier constructed variety of functions satisfying the three-term functional equations. In this article, we consider the first example among them and establish that the function is a Hecke eigen form with respect to the Hecke operators, which acts on periods. We then utilize this result to determine the action of the aforementioned operators on the derivative of the Higher order Herglotz-Zagier function. The action leads to a family of multi-term functional equations satisfied by the function.

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arXiv2026

Universal sums of generalized polygonal numbers of almost prime length

In this paper, we consider universal sums of generalized polygonal numbers. Fixing $m\in\mathbb{N}_{\geq 3}$, we show two finiteness theorems for universal sums of generalized polygonal numbers whose inputs have a restricted number $L$ of prime divisors (counting multiplicity) away from an finite set of exceptional primes. In the first theorem, we fix $m$ and uniformly bound the finite check independent of $L\geq 900$, and in the second theorem, we give an optimal bound for the finiteness check if $L$ is larger than a constant times $\log(m)$.

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arXiv2024

Sums of generalized polygonal numbers of almost prime "length"

In this paper, we consider sums of three generalized $m$-gonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restrictions on $m$ modulo $30$, we show that a density one set of integers is represented as such a sum, where the parameters are restricted to have at most 6361 prime factors. Moreover, if the squarefree part of $f_m(n)$ is sufficiently large, then $n$ is represented as such a sum, where $f_m(n)$ is a natural linear function in $n$.

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arXiv2024

Representation of even Gaussian integer à la Chen

In this article, we represent an even Gaussian integer with sufficiently large norm as a sum of a Gaussian prime and a Gaussian integer with at most two Gaussian prime factors akin to Chen in the rational case.

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arXiv2023

A note on odd zeta values over any number field and Extended Eisenstein series

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 number field extension of Eisenstein series, which satisfies the transformation $z \mapsto -1/z$ like an integral weight modular form over SL$_2(\Z)$. The results provide number of important applications, which are important in studying the behaviour of odd zeta values as well as Lambert series in an arbitrary number field.

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arXiv2023

Explicit transformations for generalized Lambert series associated with the divisor function $σ_{a}^{(N)}(n)$ and their applications

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 $σ_a^{(N)}(n)$, and is extended to a wider region by analytic continuation. For $N=1$, this Lambert series plays an important role in string theory scattering amplitudes as can be seen in the recent work of Dorigoni and Kleinschmidt. These transformations exhibit several identities - a new generalization of Ramanujan's formula for $ζ(2m+1)$, an identity associated with extended higher Herglotz functions, generalized Dedekind eta-transformation, Wigert's transformation etc., all of which are derived in this paper, thus leading to their uniform proofs. A special case of one of these explicit transformations naturally leads us to consider generalized power partitions with ``$n^{2N-1}$ copies of $n^{N}$''. Asymptotic expansion of their generating function as $q\to1^{-}$ is also derived which generalizes Wright's result on the plane partition generating function. In order to obtain these transformations, several new intermediate results are required, for example, a new reduction formula for Meijer $G$-function and an almost closed-form evaluation of $\left.\frac{\partial E_{2N, β}(z^{2N})}{\partialβ}\right|_{β=1}$, where $E_{α, β}(z)$ is a two-variable Mittag-Leffler function.

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arXiv2022

Lambert series of logarithm, the derivative of Deninger's function $R(z)$ and a mean value theorem for $ζ\left(\frac{1}{2}-it\right)ζ'\left(\frac{1}{2}+it\right)$

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 derivative of Deninger's function $R(z)$. In the course of obtaining the transformation, new important properties of $ψ_1(z)$ are derived, as is a new representation for the second derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$ evaluated at $b=1$. Our transformation readily gives the complete asymptotic expansion of $\sum\limits_{n=1}^{\infty}\displaystyle\frac{\log(n)}{e^{ny}-1}$ as $y\to0$. An application of the latter is that it gives the asymptotic expansion of $ \displaystyle\int_{0}^{\infty}ζ\left(\frac{1}{2}-it\right)ζ'\left(\frac{1}{2}+it\right)e^{-δt}\, dt$ as $δ\to0$.

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arXiv2022

On a Conjecture of Sun about sums of restricted squares

In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms $2^a3^b$ and $2^c5^d$. We obtain an ineffective generalization of results of Gauss and Legendre on sums of three squares and an effective generalization of Lagrange's four-square theorem.

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arXiv2021

Equivalent criterion for the grand Riemann hypothesis associated to Maass cusp forms

In this article, we obtain transformation formulas analogous to the identity of Ramanujan, Hardy and Littlewood in the setting of primitive Maass cusp form over the congruence subgroup $Γ_0(N)$ and also provide an equivalent criterion of the grand Riemann hypothesis for the $L$-function associated to the primitive Maass cusp form over $Γ_0(N)$.

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arXiv2021

Number field analogue of divisor function à la Koshliakov

In this article, we study a divisor function in an arbitrary number field akin to Koshliakov's work on Vorono\"{\dotlessi} summation formula. More precisely, we generalize Koshliakov's kernel and Koshliakov's transform over any number field to obtain identities for the Lambert series associated to the divisor function in an arbitrary number field.

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arXiv2021

Explicit identities on zeta values over imaginary quadratic field

In this article, we study special values of the Dedekind zeta function over an imaginary quadratic field. The values of the Dedekind zeta function at any even integer over any totally real number field is quite well known in literature. In fact, in one of the famous article, Zagier obtained an explicit formula for Dedekind zeta function at point 2 and conjectured an identity at any even values over any number field. We here exhibit the identities for both even and odd values of the Dedekind zeta function over an imaginary quadratic field which are analogous to Ramanujan's identities for even and odd zeta values over $\Q$. Moreover, any complex zeta values over imaginary quadratic field may also be evaluated from our identities.

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arXiv2020

An analogue of Wilton's formula and values of Dedekind zeta functions

J. R. Wilton obtained an expression for the product of two Riemann zeta functions. This expression played a crucial role to find the approximate functional equation for the product of two Riemann zeta functions in the critical region. We find analogous expressions for the product of two Dedekind zeta functions and then use these expressions to find some expressions for Dedekind zeta values attached to arbitrary real as well as quadratic number fields at any positive integer.

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arXiv2020

Piltz divisor problem over number fields à la Voronoï

In this article, we study the Piltz divisor problem, which is sometimes called the generalized Dirichlet divisor problem, over number fields. We establish an identity akin to Voronoï's formula concerning the error term in the Dirichlet divisor problem.

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arXiv2020

Fermat's polygonal number theorem for repeated generalized polygonal numbers

In this paper, we consider sums of generalized polygonal numbers with repeats, generalizing Fermat's polygonal number theorem which was proven by Cauchy. In particular, we obtain the minimal number of generalized $m$-gonal numbers required to represent every positive integer and we furthermore generalize this result to obtain optimal bounds when many of the generalized $m$-gonal numbers are repeated $r$ times, where $r\in\mathbb{N}$ is fixed.

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arXiv2017

A Note On Signs Of Fourier Coefficients Of Two Cusp Forms

Kohnen and Sengupta proved that two cusp forms of different integral weights with real algebraic Fourier coefficients have infinitely many Fourier coefficients of the same as well as of opposite sign, up to the action of a Galois automorphism. Recently Gun, Kohnen and Rath strengthen their result by comparing the simultaneous sign changes of Fourier coefficients of two cusp forms with arbitrary real Fourier coefficients. The simultaneous sign changes of Fourier coefficients of two same integral weight cusp forms follow from an earlier work of Ram Murty. In this note we compare the signs of the Fourier coefficients of two cusp forms simultaneously for the congruence subgroup $Γ_0(\mathit{N})$ where the coefficients lie in an arithmetic progression. Next we consider an analogous question for the particular sparse sequences of Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group.

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