arXiv subjects

Rupam Barman

Publications and source records attributed to Rupam Barman.

At least 37 records · Page 2Linked to original sources

arXiv2023

Hypergeometric functions for Dirichlet characters and Peisert-like graphs on $\mathbb{Z}_n$

For a prime $p\equiv 3\pmod 4$ and a positive integer $t$, let $q=p^{2t}$. The Peisert graph of order $q$ is the graph with vertex set $\mathbb{F}_q$ such that $ab$ is an edge if $a-b\in\langle g^4\rangle\cup g\langle g^4\rangle$, where $g$ is a primitive element of $\mathbb{F}_q$. In this paper, we construct a similar graph with vertex set as the commutative ring $\mathbb{Z}_n$ for suitable $n$, which we call \textit{Peisert-like} graph and denote by $G^\ast(n)$. Owing to the need for cyclicity of the group of units of $\mathbb{Z}_n$, we consider $n=p^α$ or $2p^α$, where $p\equiv 1\pmod 4$ is a prime and $α$ is a positive integer. For primes $p\equiv 1\pmod 8$, we compute the number of triangles in the graph $G^\ast(p^α)$ by evaluating certain character sums. Next, we study cliques of order 4 in $G^\ast(p^α)$. To find the number of cliques of order $4$ in $G^\ast(p^α)$, we first introduce hypergeometric functions containing Dirichlet characters as arguments, and then express the number of cliques of order $4$ in $G^\ast(p^α)$ in terms of these hypergeometric functions.

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arXiv2023

Diagonal hypersurfaces and elliptic curves over finite fields and hypergeometric functions

Let $D_λ^{d,k}$ denote the family of diagonal hypersurface over a finite field $\mathbb{F}_q$ given by \begin{align*} D_λ^{d,k}:X_1^d+X_2^d=λdX_1^kx_2^{d-k}, \end{align*} where $d\geq2$, $1\leq k\leq d-1$, and $\gcd(d,k)=1$. Let $\#D^{d,k}_λ$ denote the number of points on $D_λ^{d,k}$ in $\mathbb{P}^{1}(\mathbb{F}_q)$. It is easy to see that $\#D_λ^{d,k}$ is equal to the number of distinct zeros of the polynomial $y^d-dλy^k+1\in \mathbb{F}_q[y]$ in $\mathbb{F}_q$. In this article, we prove that $\#D^{d,k}_λ$ is also equal to the number of distinct zeros of the polynomial $y^{d-k}(1-y)^k-(dλ)^{-d}$ in $\mathbb{F}_q$. We express the number of distinct zeros of the polynomial $y^{d-k}(1-y)^k-(dλ)^{-d}$ in terms of a $p$-adic hypergeometric function. Next, we derive summation identities for the $p$-adic hypergeometric functions appearing in the expressions for $\#D^{d,k}_λ$. Finally, as an application of the summation identities, we prove identities for the trace of Frobenius endomorphism on certain families of elliptic curves.

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arXiv2023

Congruences for the partition function $\text{PDO}_t(n)$ modulo powers of $2$ and $3$

Lin introduced the partition function $\text{PDO}_t(n)$, which counts the total number of tagged parts over all the partitions of $n$ with designated summands in which all parts are odd. For $k\geq0$, Lin conjectured congruences for $\text{PDO}_t(8\cdot3^kn)$ and $\text{PDO}_t(12\cdot3^kn)$ modulo $3^{k+2}$. In this article, we develop a new approach to study these congruences. We study the generating functions of $\text{PDO}_t(8\cdot3^kn)$ and $\text{PDO}_t(12\cdot3^kn)$ modulo $3^{k+3}$ for certain values of $k$. We also study $\text{PDO}_t(n)$ modulo powers of $2$. We establish infinitely many congruences for $\text{PDO}_t(n)$ modulo $8$ and $32$. We prove several congruences modulo small powers of $2$ and discuss the existence of congruences modulo arbitrary powers of $2$ similar to those in Lin's conjecture. In reference to this, we also pose some problems for future work.

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arXiv2023

Parity distribution and divisibility of Mex-related partition functions

Andrews and Newman introduced the mex-function $\text{mex}_{A,a}(λ)$ for an integer partition $λ$ of a positive integer $n$ as the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $λ$. They then defined $p_{A,a}(n)$ to be the number of partitions $λ$ of $n$ satisfying $\text{mex}_{A,a}(λ)\equiv a\pmod{2A}$. They found the generating function for $p_{t,t}(n)$ and $p_{2t,t}(n)$ for any positive integer $t$, and studied their arithmetic properties for some small values of $t$. In this article, we study the partition function $p_{mt,t}(n)$ for all positive integers $m$ and $t$. We show that for sufficiently large $X$, the number of all positive integer $n\leq X$ such that $p_{mt,t}(n)$ is an even number is at least $\mathcal{O}(\sqrt{X/3})$ for all positive integers $m$ and $t$. We also prove that for sufficiently large $X$, the number of all positive integer $n\leq X$ such that $p_{mp,p}(n)$ is an odd number is at least $\mathcal{O}(\log \log X)$ for all $m\not \equiv 0\pmod{3}$ and all primes $p\equiv 1\pmod{3}$. Finally, we establish identities connecting the ordinary partition function to $p_{mt,t}(n)$.

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arXiv2023

Certain Diophantine equations and new parity results for $21$-regular partitions

For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello investigated the parity of $b_{t}(n)$ when $t\leq 28$. They discovered new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving every prime $p$ with $p\equiv 13, 17, 19, 23 \pmod{24}$. In this paper, we investigate the parity of $b_{21}(n)$ involving the primes $p$ with $p\equiv 1, 5, 7, 11 \pmod{24}$. We prove new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving the odd primes $p$ for which the Diophantine equation $8x^2+27y^2=jp$ has primitive solutions for some $j\in\left\lbrace1,4,8\right\rbrace$, and we also prove that the Dirichlet density of such primes is equal to $1/6$. Recently, Yao provided new infinite families of congruences modulo $2$ for $b_{3}(n)$ and those congruences involve every prime $p\geq 5$ based on Newman's results. Following a similar approach, we prove new infinite families of congruences modulo $2$ for $b_{21}(n)$, and these congruences imply that $b_{21}(n)$ is odd infinitely often.

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arXiv2023

Cliques of orders three and four in the Paley-type graphs

Let $n=2^s p_{1}^{α_{1}}\cdots p_{k}^{α_{k}}$, where $s=0$ or $1$, $α_i\geq 1$, and the distinct primes $p_i$ satisfy $p_i\equiv 1\pmod{4}$ for all $i=1, \ldots, k$. Let $\mathbb{Z}_n^\ast$ denote the group of units in the commutative ring $\mathbb{Z}_n$. Recently, we defined a Paley-type graph $G_n$ of order $n$ as the graph whose vertex set is $\mathbb{Z}_n$ and $xy$ is an edge if $x-y\equiv a^2\pmod n$ for some $a\in\mathbb{Z}_n^\ast$. The Paley-type graph $G_n$ resembles the classical Paley graph in a number of ways, and adds to the list of generalizations of the Paley graph. Computing the number of cliques of a particular order in a Paley graph or its generalizations has been of considerable interest. For primes $p\equiv 1\pmod 4$ and $α\geq 1$, by evaluating certain character sums, we found the number of cliques of order $3$ in $G_{p^α}$ and expressed the number of cliques of order $4$ in $G_{p^α}$ in terms of Jacobi sums. In this article we give combinatorial proofs and find the number of cliques of orders $3$ and $4$ in $G_n$ for all $n$ for which the graph is defined.

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arXiv2022

$p$-Adic quotient sets: linear recurrence sequences

Let $(x_n)_{n\geq0}$ be a linear recurrence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In [`The quotient set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_p$', \emph{Bull. Aust. Math. Soc.} \textbf{96} (2017), 24-29], Sanna posed an open question to classify primes $p$ for which the quotient set of $(x_n)_{n\geq0}$ is dense in $\mathbb{Q}_p$. In this article, we find a sufficient condition for denseness of the quotient set of the $k$th-order linear recurrence $(x_n)_{n\geq0}$ satisfying $ x_{n}=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$ for all integers $n\geq k$ with initial values $x_0=\dots=x_{k-2}=0,x_{k-1}=1$, where $a_1,\dots,a_k\in \mathbb{Z}$ and $a_k=1$. We show that given a prime $p$, there exist infinitely many recurrence sequences of order $k\geq 2$ so that their quotient sets are not dense in $\mathbb{Q}_p$. We also study the quotient sets of linear recurrence sequences with coefficients in some arithmetic and geometric progressions.

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arXiv2022

Number of $\mathbb{F}_q$-points on Diagonal hypersurfaces and hypergeometric function

Let $D_λ^d$ denote the family of monomial deformations of diagonal hypersurface over a finite field $\mathbb{F}_q$ given by \begin{align*} D_λ^d: X_1^d+X_2^d+\cdots+X_n^d=λd X_1^{h_1}X_2^{h_2}\cdots X_n^{h_n}, \end{align*} where $d,n\geq2$, $h_i\geq1$, $\sum_{i=1}^n h_i=d$, and $\gcd(d,h_1,h_2,\ldots, h_n)=1$. The Dwork hypersurface is the case when $d=n$, that is, $h_1=h_2=\cdots =h_n=1$. Formulas for the number of $\mathbb{F}_q$-points on the Dwork hypersurfaces in terms of McCarthy's $p$-adic hypergeometric functions are known. In this article we provide a formula for the number of $\mathbb{F}_q$-points on $D_λ^d$ in terms of McCarthy's $p$-adic hypergeometric function which holds for $d\geq n$.

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arXiv2022

Arithmetic properties of certain $t$-regular partitions

For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_9(n)$ and $b_{19}(n)$. We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of $b_9(n)$ and $b_{19}(n)$ modulo $2$. We also relate $b_{t}(n)$ to the ordinary partition function, and prove that $b_{t}(n)$ satisfies the Ramanujan's famous congruences for some infinite families of $t$. For $t\in \{6,10,14,15,18,20,22,26,27,28\}$, Keith and Zanello conjectured that there are no integers $A>0$ and $B\geq 0$ for which $b_t(An+ B)\equiv 0\pmod 2$ for all $n\geq 0$. We prove that, for any $t\geq 2$ and prime $\ell$, there are infinitely many arithmetic progressions $An+B$ for which $\sum_{n=0}^{\infty}b_t(An+B)q^n\not\equiv0 \pmod{\ell}$. Next, we obtain quantitative estimates for the distributions of $b_{6}(n), b_{10}(n)$ and $b_{14}(n)$ modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and $13$-regular partition functions.

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arXiv2022

On denseness of certain direction and generalized direction sets

Direction sets, recently introduced by Leonetti and Sanna, are generalization of ratio sets of subsets of positive integers. In this article, we generalize the notion of direction sets and define {\it $k$-generalized direction sets} and {\it distinct $k$-generalized direction sets} for subsets of positive integers. We prove a necessary condition for a subset of $\mathcal{S}^{k - 1} := \{\underline{x} \in [0,1]^{k} : ||\underline{x}|| = 1\}$ to be realized as the set of accumulation points of a distinct $k$-generalized direction set. We provide sufficient conditions for some particular subsets of positive integers so that the corresponding $k$-generalized direction sets are dense in $\mathcal{S}^{k - 1}$. We also consider the denseness properties of certain direction sets and give a partial answer to a question posed by Leonetti and Sanna. Finally we consider a similar question in the framework of an algebraic number field.

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arXiv2022

Number of complete subgraphs of Peisert graphs and finite field hypergeometric functions

For a prime $p\equiv 3\pmod{4}$ and a positive integer $t$, let $q=p^{2t}$. Let $g$ be a primitive element of the finite field $\mathbb{F}_q$. The Peisert graph $P^\ast(q)$ is defined as the graph with vertex set $\mathbb{F}_q$ where $ab$ is an edge if and only if $a-b\in\langle g^4\rangle \cup g\langle g^4\rangle$. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in $P^\ast(q)$. We also give a new proof for the number of complete subgraphs of order three contained in $P^\ast(q)$ by evaluating certain character sums. The computations for the number of complete subgraphs of order four are quite tedious, so we further give an asymptotic result for the number of complete subgraphs of any order $m$ in Peisert graphs.

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arXiv2022

Certain transformations and values of p-adic hypergeometric functions

We prove two transformations for the $p$-adic hypergeometric functions which can be described as $p$-adic analogues of a Euler's transformation and a transformation of Clausen. We first evaluate certain character sums, and then relate them to the $p$-adic hypergeometric functions to deduce the transformations. We use a character sum identity proved by Ahlgren, Ono, and Penniston to deduce the $p$-adic Clausen's transformation. We also deduce special values of certain $p$-adic hypergeometric functions.

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arXiv2022

Proofs of some conjectures of Keith and Zanello on $t$-regular partition

For a positive integer $t$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity results modulo $2$ for $b_{t}(n)$ for certain values of $t$. Further, they proposed some conjectures on self-similarities of $b_t(n)$ modulo $2$ for certain values of $t$. In this paper, we prove their conjectures on $b_3(n)$ and $b_{25}(n)$. We also prove a self-similarity result for $b_{21}(n)$ modulo $2$.

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arXiv2022

$p$-Adic quotient sets: diagonal forms

For a set of integers $A$, we consider $R(A)=\{a/b: a, b\in A, b\neq 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of nonzero values attained by an integral form. This problem has been answered for quadratic forms. Very recently, Antony and Barman have studied this problem for the diagonal binary cubic forms $ax^3+by^3$, where $a$ and $b$ are integers. In this article, we study this problem for diagonal forms. We extend the results of Antony and Barman to the diagonal binary forms $ax^n+by^n$ for all $n\geq 3$. We also study $p$-adic denseness of quotients of nonzero values attained by diagonal forms of degree $n\geq 3$, where $\gcd(n,p(p-1))=1$.

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arXiv2021

Divisibility of certain $\ell$-regular partitions by $2$

For a positive integer $\ell$, let $b_{\ell}(n)$ denote the number of $\ell$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_3(n)$ and $b_{21}(n)$. We prove a specific case of a conjecture of Keith and Zanello on self-similarities of $b_3(n)$ modulo $2$. We next prove that the series $\sum_{n=0}^{\infty}b_9(2n+1)q^n$ is lacunary modulo arbitrary powers of $2$. We also prove that the series $\sum_{n=0}^{\infty}b_9(4n)q^n$ is lacunary modulo $2$.

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arXiv2021

Proof of some conjectural congruences of da Silva and Sellers

Let $p_{\{3, 3\}}(n)$ denote the number of $3$-regular partitions in three colours. In a very recent paper, da Silva and Sellers studied certain arithmetic properties of $p_{\{3, 3\}}(n)$. They further conjectured four Ramanujan-like congruences modulo $5$ satisfied by $p_{\{3, 3\}}(n)$. In this article, we confirm the conjectural congruences of da Silva and Sellers using the theory of modular forms.

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arXiv2021

$p$-adic quotient sets: Cubic forms

For $A\subseteq \{1, 2, \ldots\}$, we consider $R(A)=\{a/b: a, b\in A\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of nonzero values assumed by a cubic form. We study this problem for the cubic forms $ax^3+by^3$, where $a$ and $b$ are integers. We also prove that if $A$ is the set of nonzero values assumed by a non-degenerate, integral and primitive cubic form with more than 9 variables, then $R(A)$ is dense in $\mathbb{Q}_p$.

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arXiv2021

On a Paley-type graph on $\mathbb{Z}_n$

Let $q$ be a prime power such that $q\equiv 1\pmod{4}$. The Paley graph of order $q$ is the graph with vertex set as the finite field $\mathbb{F}_q$ and edges defined as, $ab$ is an edge if and only if $a-b$ is a non-zero square in $\mathbb{F}_q$. We attempt to construct a similar graph of order $n$, where $n\in\mathbb{N}$. For suitable $n$, we construct the graph where the vertex set is the finite commutative ring $\mathbb{Z}_n$ and edges defined as, $ab$ is an edge if and only if $a-b\equiv x^2\pmod{n}$ for some unit $x$ of $\mathbb{Z}_n$. We look at some properties of this graph. For primes $p\equiv 1\pmod{4}$, Evans, Pulham and Sheehan computed the number of complete subgraphs of order 4 in the Paley graph. Very recently, Dawsey and McCarthy find the number of complete subgraphs of order 4 in the generalized Paley graph of order $q$. In this article, for primes $p\equiv 1\pmod{4}$ and any positive integer $α$, we find the number of complete subgraphs of order 3 and 4 in our graph defined over $\mathbb{Z}_{p^α}$.

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