arXiv subjects
Publications and source records attributed to Jaban Meher.
We settle a conjecture proposed by B. Maji and T. Sarkar regarding the location of zeros of a two-parameter family of reciprocal polynomials, $R_{k,\ell}(z)$ for positive integers $k$ and $\ell$. These polynomials are generalizations of Ramanujan polynomials studied by M. R. Murty, C. Smyth, and R. Wang. More specifically, we show that except for two real zeros, all other zeros of $R_{k,\ell}(z)$ lie on the unit circle.
In this article, we distinguish Siegel cuspidal eigenforms of degree two on the full symplectic group from the signs of their Hecke eigenvalues. To establish our theorem, we obtain a result towards simultaneous sign changes of eigenvalues of two Siegel eigenforms. In the course of the proof, we also prove that the Satake $p$-parameters of two different Siegel eigenforms are distinct for a set of primes $p$ of density 1. The main ingredient to prove the latter result is the theory of Galois representations attached to Siegel eigenforms.
In this paper we first prove an isomorphism between certain spaces of Jacobi forms. Using this isomorphism, we study the mod $p$ theory of Hermitian Jacobi forms over $\mathbb{Q}(i)$. We then apply the mod $p$ theory of Hermitian Jacobi forms to characterize $U(p)$ congruences and to study Ramanujan-type congruences for Hermitian Jacobi forms and Hermitian modular forms of degree $2$ over $\mathbb{Q}(i)$.
We study the signs of the Fourier coefficients of a newform. Let $f$ be a normalized newform of weight $k$ for $Γ_0(N)$. Let $a_f(n)$ be the $n$th Fourier coefficient of $f$. For any fixed positive integer $m$, we study the distribution of the signs of $\{a_f(p^m)\}_p$, where $p$ runs over all prime numbers. We also find out the abscissas of absolute convergence of two Dirichlet series with coefficients involving the Fourier coefficients of cusp forms and the coefficients of symmetric power $L$-functions.
For $m\ge 2$, let $π$ be an irreducible cuspidal automorphic representation of $GL_m(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. Let $a_π(n)$ be the $n^{th}$ coefficient of the $L$-function attached to $π$. Goldfeld and Sengupta have recently obtained a bound for $\sum_{n\le x} a_π(n)$ as $x \rightarrow \infty$. For $m\ge 3$ and $π$ not a symmetric power of a $GL_2(\mathbb{A}_{\mathbb{Q}})$-cuspidal automorphic representation with not all finite primes unramified for $π$, their bound is better than all previous bounds. In this paper, we further improve the bound of Golfeld and Sengupta. We also prove a quantitative result for the number of sign changes of the coefficients of certain automorphic $L$-functions, provided the coefficients are real numbers.
In this paper, we give criteria for infinitely many sign changes of the coefficients of any Dirichlet series if the coefficients are real numbers. We also provide examples where our criteria are applicable.
In this paper, we present a quantitative result for the number of sign changes for the sequences $\{a(n^j)\}_{n\ge 1}, j=2,3,4$ of the Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group $SL_2(\mathbb{Z})$. We also prove a similar kind of quantitative result for the number of sign changes of the $q$-exponents $c(p) (p {vary over primes})$ of certain generalized modular functions for the congruence subgroup $Γ_0(N)$, where $N$ is square-free.
We investigate the cases for which products of two quasimodular or nearly holomorphic eigenforms are eigenforms. We also genaralize the results of Ghate \cite{ghate1} to the case of Rankin-Cohen brackets.