arXiv subjects

Parth Chavan

Publications and source records attributed to Parth Chavan.

5 recordsLinked to original sources

arXiv2025

The joy of multisection: elementary approaches to Ramanujan's lacunary identities for Bernoulli numbers

An identity by Ramanujan related to the multisection of Bernoulli numbers is revisited. Two alternative approaches are proposed, both relying on the multisection technique. A geometric approach reveals the role played by the symmetries of the summation domain in the complex plane induced by the multisection technique. The second approach, based on generating functions, allows us to extend Ramanujan's identity to other special functions such as Eisenstein's series.

math.NT↗

arXiv2023

Counting Ideals in Numerical Semigroups

If $S$ is a numerical semigroup, let $m(S,k)$ denote the number of ideals of $S$ with codimension $k$ and let $n(S,k)$ denote the number of ideals of $S$ with conductor $k$. We compute the generating function of the sequence $m(S,k)$ for all numerical semigroups of embedding dimension $2$ and for $S = \langle 3,n+2,2n+1\rangle$. We also prove that the sequence $n(S,k)$ becomes stationary after a certain term and compute the stationary terms for numerical semigroups of the form $\langle a,a+1 \rangle$.

math.NT↗

arXiv2022

Hurwitz Zeta Functions and Ramanujan's Identity for Odd Zeta Values

Inspired by a famous formula of Ramanujan for odd zeta values, we prove an analogous formula involving the Hurwitz zeta function. We introduce a new integral kernel related to the Hurwitz zeta function, generalizing the integral kernel associated to Ramanujan's identity. We also derive several infinite families of identities analogous to Ramanujan's formula.

math.NT↗

arXiv2022

Hypergraph Fuss-Catalan Numbers

The Catalan numbers $C_n$ are an extremely well-studied sequence of numbers that appear as the answer to many combinatorial problems. Two generalizations of these numbers that have been studied are the Fuss-Catalan numbers and the Hypergraph Catalan numbers. In this paper, we study the combination of these, the Hypergraph Fuss-Catalan numbers. We provide some combinatorial interpretations of these numbers, as well as describe their generating function.

math.CO↗

arXiv2021

Dirichlet Series Under Standard Convolutions: Variations on Ramanujan's Identity for Odd Zeta Values

Inspired by a famous identity of Ramanujan, we propose a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; its specialization produces new identities and recovers several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by A. Dixit and collaborators, or the convolution of Bernoulli numbers by Y. Komori and collaborators.

math.NT↗