arXiv subjects

Tapas Chatterjee

Publications and source records attributed to Tapas Chatterjee.

At least 19 recordsLinked to original sources

arXiv2026

Quadratic Residue Codes over $\mathbb{Z}_{121}$

In this paper, we construct a special family of cyclic codes, known as quadratic residue codes of prime length $ p \equiv \pm 1 \pmod{44} ,$ $ p \equiv \pm 5 \pmod{44} ,$ $ p \equiv \pm 7 \pmod{44} ,$ $ p \equiv \pm 9 \pmod{44} $ and $ p \equiv \pm 19 \pmod{44} $ over $\mathbb{Z}_{121}$ by defining them using their generating idempotents. Furthermore, the properties of these codes and extended quadratic residue codes over $\mathbb{Z}_{121}$ are discussed, followed by their Gray images. Also, we show that the extended quadratic residue code over $\mathbb{Z}_{121}$ possesses a large permutation automorphism group generated by shifts, multipliers, and inversion, making permutation decoding feasible. As examples, we construct new codes with parameters $[55,5,33]$ and $[77,7,44].$

cs.IT↗

arXiv2026

On transcendence of non-periodic continued fractions associated with modular forms and arithmetic functions

The purpose of this article is two-folds. Firstly, we establish two sufficient conditions under which the sequence $\{f(n)\pmod{m}: n\geq1\}$ is non-periodic, where $f(n)$ is an arithmetic function. As consequences, we deduce that the sequences associated with the Ramanujan tau function $\tau(n)$ as well as the Fourier coefficients of certain normalized Eisenstein series $E_k(z)$ modulo $m$ are non-periodic. Further, we deduce that the sequence arising from Nathanson's totient function $\Phi(n)$, the classical Euler's totient function $\varphi(n)$, sum of divisor function $\sigma(n)$, their Dirichlet convolution $\sigma*\varphi(n)$, Jordan's totient function $J_k(n)$, and unitary totient function $\varphi^*(n)$ modulo $m$, are non-periodic for certain modulo $m$. In addition, we extend a result of Ayad and Kihel \cite{r1} on the non-periodicity of certain arithmetic function $g(n)$. On the other hand, we construct several transcendental numbers arising from the continued fractions attached with $\tau(n)$, $E_k(z)$, $\Phi(n)$, $g(n)$, $\varphi(n)$, $\sigma(n)$, $\sigma*\varphi(n)$, $J_k(n)$, and $\varphi^*(n)$.

math.GM↗

arXiv2025

Distribution of Farey fractions with $k$-free denominators

We investigate the distributional properties of the sequence of Farey fractions with $k$-free denominators in residue classes, defined as \[\mathscr{F}_{Q,k}^{(m)}:=\left\{\frac{a}{q}\ |\ 1\leq a\leq q\leq Q,\ \gcd(a,q)=1,\ q\ \text{is}\ k\text{-free}\ \&\ q\equiv b\pmod{m} \right\}.\] We show that $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$ is equidistributed modulo one, and prove analogues of the classical results of Franel, Landau, and Niederreiter for $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$, particularly, deriving an equivalent form of the generalized Riemann hypothesis (GRH) for Dirichlet $L$-functions in terms of the distribution of $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$. Beyond examining the global distribution, we also study the local statistics of these sequences. We establish formulas for all levels ($\nu\ge 2$) of correlation measure. Specifically, we show the existence of the limiting pair ($\nu=2$) correlation function and provide an explicit expression for it. Our results are based upon the estimation of weighted Weyl sums and weighted lattice point counting in restricted domains.

math.NT↗

arXiv2024

On characterization of Monogenic number fields associated with certain quadrinomials and its applications

Let $f(x)=x^{n}+ax^{3}+bx+c$ be the minimal polynomial of an algebraic integer $\theta$ over the rationals with certain conditions on $a,~b,~c,$ and $n.$ Let $K=\mathbb{Q}(\theta)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $\theta.$ As an interesting result, we establish necessary and sufficient conditions for the field $K=\mathbb{Q}(\theta)$ to be monogenic. Finally, we investigate the types of solutions to certain differential equations associated with the polynomial $f(x).$

math.NT↗

arXiv2024

On characterization of prime divisors of the index of a quadrinomial

Let $\theta$ be an algebraic integer and $f(x)=x^{n}+ax^{n-1}+bx+c$ be the minimal polynomial of $\theta$ over the rationals. Let $K=\mathbb{Q}(\theta)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $f(x).$ As a fascinating corollary, we deduce necessary and sufficient conditions for the monogenity of the field $K=\mathbb{Q}(\theta),$ where $\theta$ is associated with certain quadrinomials.

math.NT↗

arXiv2024

Special values of derivatives of certain $L$-functions

In this paper we address the question of non-vanishing of $L'(0,f)$ where $f$ is an algebraic valued periodic function. In 2011, Gun, Murty and Rath studied the nature of special values of the derivatives of even Dirichlet-type functions and proved that it can be either zero or transcendental. Here for some special cases we characterize the set of functions for which $L'(0,f)$ is zero or transcendental. Using a theorem of Ramachandra about multiplicative independence of cyclotomic units we also provide some non-trivial examples of functions where $L'(0,f)$ is zero. Finally, assuming Schanuel's conjecture we derive the algebraic independence of special values of derivatives of $L$-functions.

math.NT↗

arXiv2024

On Characterizing Potential Friends of 20

Does $20$ have a friend? Or is it a solitary number? A folklore conjecture asserts that $20$ has no friends i.e. it is a solitary number. In this article, we prove that, a friend $N$ of $20$ is of the form $N=2\cdot5^{2a}\cdot m^2$, with $(3,m)=(7,m)=1$ and it has at least six distinct prime divisors. Furthermore, we show that $\Omega(N)\geq 2\omega(N)+6a-5$ and if $\Omega(m)\leq K$ then $N< 10\cdot 6^{(2^{K-2a+3}-1)^2}$, where $\Omega(n)$ and $\omega(n)$ denote the total number of prime divisors and the number of distinct prime divisors of the integer $n$ respectively. In addition, we deduce that, not all exponents of odd prime divisors of friend $N$ of $20$ are congruent to $-1$ modulo $f$, where $f$ is the order of $5$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ such that $3\mid f$ and $p$ is a prime congruent to $1$ modulo $6$. Also, we prove necessary upper bounds for all prime divisors of friends of 20 in terms of the number of divisors of the friend. In addition, we prove that, if $P$ is the largest prime divisor of $N$ then $P<N^{\frac{1}{4}}$.

math.GM↗

arXiv2024

On MDS Property of g-Circulant Matrices

Circulant Maximum Distance Separable (MDS) matrices have gained significant importance due to their applications in the diffusion layer of the AES block cipher. In $2013$, Gupta and Ray established that circulant involutory matrices of order greater than $3$ cannot be MDS over $\mathbb{F}_{2^m}$. This finding prompted a generalization of circulant matrices and the involutory property of matrices by various authors. In $2016$, Liu and Sim introduced cyclic matrices by changing the permutation of circulant matrices. In $1961,$ Friedman introduced $g$-circulant matrices which form a subclass of cyclic matrices. In this article, we first discuss $g$-circulant matrices with involutory and MDS properties. We prove that $g$-circulant involutory matrices of order $k \times k$ cannot be MDS unless $g \equiv -1 \pmod k.$ Next, we delve into $g$-circulant semi-involutory and semi-orthogonal matrices with entries from finite fields. We establish that the $k$-th power of the associated diagonal matrices of a $g$-circulant semi-orthogonal (semi-involutory) matrix of order $k \times k$ results in a scalar matrix. These findings extend the recent results on circulant matrices established by Kumar {\it{et al.}} $(2026)$ and Chatterjee {\it{et al.}} $(2022)$. Furthermore, we prove that cyclic matrices of order $2^{d} \times 2^{d}$ over finite fields of characteristic $2$ cannot simultaneously possess both the MDS and semi-orthogonal properties.

cs.CR↗

arXiv2024

A note on MDS Property of Circulant Matrices

In $2014$, Gupta and Ray proved that the circulant involutory matrices over the finite field $\mathbb{F}_{2^m}$ can not be maximum distance separable (MDS). This non-existence also extends to circulant orthogonal matrices of order $2^d \times 2^d$ over finite fields of characteristic $2$. These findings inspired many authors to generalize the circulant property for constructing lightweight MDS matrices with practical applications in mind. Recently, in $2022,$ Chatterjee and Laha initiated a study of circulant matrices by considering semi-involutory and semi-orthogonal properties. Expanding on their work, this article delves into circulant matrices possessing these characteristics over the finite field $\mathbb{F}_{2^m}.$ Notably, we establish a correlation between the trace of associated diagonal matrices and the MDS property of the matrix. We prove that this correlation holds true for even order semi-orthogonal matrices and semi-involutory matrices of all orders. Additionally, we provide examples that for circulant, semi-orthogonal matrices of odd orders over a finite field with characteristic $2$, the trace of associated diagonal matrices may possess non-zero values.

cs.CR↗

arXiv2024

A note on cyclic MDS and non-MDS matrices

In $1998,$ Daemen {\it{ et al.}} introduced a circulant Maximum Distance Separable (MDS) matrix in the diffusion layer of the Rijndael block cipher, drawing significant attention to circulant MDS matrices. This block cipher is now universally acclaimed as the AES block cipher. In $2016,$ Liu and Sim introduced cyclic matrices by modifying the permutation of circulant matrices and established the existence of MDS property for orthogonal left-circulant matrices, a notable subclass within cyclic matrices. While circulant matrices have been well-studied in the literature, the properties of cyclic matrices are not. Back in $1961$, Friedman introduced $g$-circulant matrices which form a subclass of cyclic matrices. In this article, we first establish a permutation equivalence between a cyclic matrix and a circulant matrix. We explore properties of cyclic matrices similar to $g$-circulant matrices. Additionally, we determine the determinant of $g$-circulant matrices of order $2^d \times 2^d$ and prove that they cannot be simultaneously orthogonal and MDS over a finite field of characteristic $2$. Furthermore, we prove that this result holds for any cyclic matrix.

cs.CR↗

arXiv2024

A Characterization of Semi-Involutory MDS Matrices

In symmetric cryptography, maximum distance separable (MDS) matrices with computationally simple inverses have wide applications. Many block ciphers like AES, SQUARE, SHARK, and hash functions like PHOTON use an MDS matrix in the diffusion layer. In this article, we first characterize all $3 \times 3$ irreducible semi-involutory matrices over the finite field of characteristic $2$. Using this matrix characterization, we provide a necessary and sufficient condition to construct MDS semi-involutory matrices using only their diagonal entries and the entries of an associated diagonal matrix. Finally, we count the number of $3 \times 3$ semi-involutory MDS matrices over any finite field of characteristic $2$.

cs.CR↗

arXiv2024

Algebraic identities among $q$- analogue of Euler double zeta values

In 2003, Zudilin presented a $q$-analogue of Euler's identity for one of the variants of $q$-double zeta function. This article focuses on exploring identities related to another variant of $q$-double zeta function and its star variant. Using a $q$-analogue of the Nielsen Reflexion Formula for $q>1$, we investigate identities involving different versions of $q$-analogues of the Riemann zeta function and the double-zeta function. Additionally, we analyze the behavior of $\zeta_q(s_1, s_2)$ as $s_1$ and $s_2$ approach to $0$ and compare these limits to those of the classical double-zeta function. Finally, we discuss the $q$-analogue of the Mordell-Tornheim $r$-ple zeta function and its relation with the $q$-double zeta function.

math.NT↗

arXiv2024

Transcendental nature of $p$-adic digamma values

For a fixed prime $p$, Murty and Saradha (2008) studied the transcendental nature of special values of the $p$-adic digamma function, denoted as $\psi_p(r/p)+ \gamma_p$. This research was later extended by Chatterjee and Gun in 2014, who investigated the case of $\psi_p(r/p^n)+ \gamma_p$, for any integer $n>1$. In this article, we generalize their results for distinct prime powers and explore the transcendental nature of the $p$-adic digamma values, with at most one exception. Further, we investigate the multiplicative independence of cyclotomic numbers satisfying certain conditions. Using this, we prove the transcendental nature of $p$-adic digamma values corresponding to $\psi_p(r/pq)+ \gamma_p$, where $p, q$ are distinct primes.

math.NT↗

arXiv2024

On arithmetic nature of $q$-analogue of the generalized Stieltjes constants

In this article, our aim is to extend the research conducted by Kurokawa and Wakayama in 2003, particularly focusing on the $q$-analogue of the Hurwitz zeta function. Our specific emphasis lies in exploring the coefficients in the Laurent series expansion of a $q$-analogue of the Hurwitz zeta function around $s=1$. We establish the closed-form expressions for the first two coefficients in the Laurent series of the $q$-Hurwitz zeta function. Additionally, utilizing the reflection formula for the digamma function and the identity of Bernoulli polynomials, we explore transcendence results related to $\gamma_0(q,x)$ for $q>1$ and $0 < x <1$, where $\gamma_0(q,x)$ is the constant term which appears in the Laurent series expansion of $q$-Hurwitz zeta function around $s=1$. Furthermore, we put forth a conjecture about the linear independence of special values of $\gamma_0(q,x)$ along with $1$ at rational arguments with co-prime conditions, over the field of rational numbers. Finally, we show that at least one more than half of the numbers are linearly independent over the field of rationals.

math.NT↗

arXiv2024

Linear independence of $q$-analogue of the generalized Stieltjes constants over number fields

In this article, we aim to extend the research conducted by Chatterjee and Garg in 2024, particularly focusing on the $q$-analogue of the generalized Stieltjes constants. These constants constitute the coefficients in the Laurent series expansion of a $q$-analogue of the Hurwitz zeta function around $s=1$. Chatterjee and Garg previously established arithmetic results related to $\gamma_0(q,x)$, for $q>1$ and $0 < x <1$ over the field of rational numbers. Here, we broaden their findings to encompass number fields $\mathbb{F}$ in two scenarios: firstly, when $\mathbb{F}$ is linearly disjoint from the cyclotomic field $\mathbb{Q}(\zeta_b)$, and secondly, when $\mathbb{F}$ has non-trivial intersection with $\mathbb{Q}(\zeta_b)$, with $b \geq 3$ being any positive integer.

math.NT↗

arXiv2024

A note on necessary conditions for a friend of 10

Solitary numbers are shrouded with mystery. A folklore conjecture assert that 10 is a solitary number i.e. it has no friends. In this article, we establish that if $N$ is a friend of $10$ then it must be odd square with at least seven distinct prime factors, with $5$ being the least one. Moreover there exists a prime factor $p$ of $N$ such that $2a+1\equiv 0 \pmod f$ and $5^{f}\equiv 1 \pmod p$ where $f$ is the smallest odd positive integer greater than $1$ and less than or equal to $\min\{ 2a+1,p-1\}$, provided $5^{2a}\mid \mid N$. Further, there exist prime factors $p$ and $q$ (not necessarily distinct) of $N$ such that $p\equiv1 \pmod {10}$ and $q\equiv 1\pmod 6$. Besides, we prove that if a Fermat prime $F_k$ divides $N$ then $N$ must have a prime factor congruent to $1$ modulo $2F_k$. Also, if we consider the form of $N$ as $N=5^{2a}m^2$ then $m$ is non square-free. Furthermore, we show that $\Omega(N)\geq 2\omega(N)+6a-4$ and if $\Omega(m)\leq K$ then $N< 5\cdot 6^{(2^{K-2a+1}-1)^2}$ where $\Omega(n)$ and $\omega(n)$ denote the total number of prime factors and the number of distinct prime factors of the integer $n$ respectively.

math.NT↗

arXiv2019

Shifted Euler constants and a generalization of Euler-Stieltjes constants

The purpose of this article is twofold. First, we introduce the constants $\zeta_k(\alpha,r,q)$ where $\alpha \in (0,1)$ and study them along the lines of work done on Euler constant in arithmetic progression $\gamma(r,q)$ by Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants are used for evaluation of certain integrals involving error term for Dirichlet divisor problem with congruence conditions and also to provide a closed form expression for the value of a class of Dirichlet L-series at any real critical point. In the second half of this paper, we consider the behaviour of the Laurent Stieltjes constants $\gamma_k(\chi)$ for a principal character $\chi.$ In particular, we study a generalization of the "Generalized Euler constants" introduced by Diamond and Ford in 2008. We conclude with a short proof for a closed form expression for the first generalized Stieltjes constant $\gamma_1(r/q)$ which was given by Blagouchine in 2015.

math.NT↗

arXiv2018

Linear Independence of Harmonic Numbers over the field of Algebraic Numbers

Let $H_n =\sum\limits_{k=1}^n \frac{1}{k}$ be the $n$-th harmonic number. Euler extended it to complex arguments and defined $H_r$ for any complex number $r$ except for the negative integers. In this paper, we give a new proof of the transcendental nature of $H_r$ for rational $r$. For some special values of $q>1,$ we give an upper bound for the number of linearly independent harmonic numbers $H_{a/q}$ with $ 1 \leq a \leq q$ over the field of algebraic numbers. Also, for any finite set of odd primes $J$ with $|J|=n,$ define $$W_J=\overline{\mathbb{Q}}-\text {span of } \{ H_1, \ H_{a_{j_i}/q_i} | \ 1 \leq a_{j_i} \leq q_i -1, \ 1 \leq j_i \leq q_i-1, \ \ \forall q_i \in J\}.$$ Finally, we show that $$\text{ dim }_{\overline{\mathbb{Q}}} ~W_J=\sum\limits_{\substack{i=1 \\ q_i \in J}}^n \frac{\phi (q_i )}{2} + 2.$$

math.NT↗