Searcharxiv⌕ Search

arXiv subjects

Srinivas Kotyada

Publications and source records attributed to Srinivas Kotyada.

9 recordsLinked to original sources
arXiv2025

Oscillations and first-ever negative Fourier coefficients of symmetric square L-functions over sparse set

Let $sym^{2} f$ denote the symmetric square lift of a Hecke eigenform $f \in S_{k}(Γ_{0}(N))$ with the $n^{\rm th}$-Fourier coefficients $ λ_{sym^{2}f}(n)$. In this article, we prove an estimate for the first moment of the sequence $\{ λ_{sym^{2}f}(\mathcal{Q}(\underline{x}))\}_{\mathcal{Q} \in \mathcal{S}_{D}, \underline{x} \in \mathbb{Z}^{2}}$ where $\mathcal{S}_{D}$ denotes the set of in-equivalent reduced forms of the discriminant $D$. More precisely, we establish an estimate for the following sum: \begin{equation*} \begin{split} S(sym^{2}f, D; X ) &= \sideset{}{^{\flat }}\sum_{\substack{\mathcal{Q}(\underline{x}) \leq X \\ \underline{x} \in \mathbb{Z}^{2} ,~ \mathcal{Q} \in \mathcal{S}_{D} \\ \gcd(\mathcal{Q}(\underline{x}),N) =1 }} λ_{sym^{2}f}(\mathcal{Q}(\underline{x})), \end{split} \end{equation*} Moreover, we consider a question concerning the behavior of signs of the Fourier coefficients $λ_{sym^{2}f}(n),$ supported on the set of integers represented by reduced forms of the discriminant $D$. We determine the size of $n_{sym^{2}f, D}$ (see definition before \thmref{ExtMatKLSW}), in terms of the conductor of the associated $L$-functions.

math.NT↗
arXiv2023

A short note on number fields defined by exponential Taylor polynomials

Let $n$ be a positive integer and $f_n(x)= 1+x+\frac{x^2}{2!}+\cdots + \frac{x^n}{n!}$ denote the $n$-th Taylor polynomial of the exponential function. Let $K = \mathbf{Q}(θ)$ be an algebraic number field where $θ$ is a root of $f_n(x)$ and $\mathbf{Z}_K$ denote the ring of algebraic integers of $K$. In this paper, we prove that for any prime $p$, $p$ does not divide the index of the subgroup $\mathbf{Z}[θ]$ in $\mathbf{Z}_K$ if and only if $p^2\nmid n!$.

math.NT↗
arXiv2023

On the irreducibility of extended Laguerre Polynomials

Let $m\geq 1$ and $a_m$ be integers. Let $α$ be a rational number which is not a negative integer such that $α= \frac{u}{v}$ with $\gcd(u,v) = 1, v>0$. Let $ϕ(x)$ belonging to $\Z[x]$ be a monic polynomial which is irreducible modulo all the primes less than or equal to $vm+u$. Let $a_i(x)$ with $0\leq i\leq m-1$ belonging to $\Z[x]$ be polynomials having degree less than $\degϕ(x)$. Assume that the content of $(a_ma_0(x))$ is not divisible by any prime less than or equal to $vm+u$. In this paper, we prove that the polynomials $L_{m,α}^ϕ(x) = \frac{1}{m!}(a_mϕ(x)^m+\sum\limits_{j=0}^{m-1}b_ja_j(x)ϕ(x)^j)$ are irreducible over the rationals for all but finitely many $m$, where $b_j = \binom{m}{j}(m+α)(m-1+α)\cdots (j+1+α)~~~\mbox{ for }0\leq j\leq m-1$. Further, we show that $L_{m,α}^ϕ(x)$ is irreducible over rationals for each $α\in \{0, 1, 2, 3, 4\}$ unless $(m, α) \in \{ (1,0), (2,2), (4,4),(6,4)\}.$ For proving our results, we use the notion of $ϕ$-Newton polygon and some results from analytic number theory. We illustrate our results through examples.

math.NT↗
arXiv2017

A note on Euclidean cyclic cubic fields

Let $K$ be a cyclic cubic field and $\mathcal{O}_K$ be its ring of integers. In this note we prove that all cyclic cubic number fields with conductors in the interval $ [73, 11971]$ and with class number one are Euclidean.

math.NT↗
arXiv2017

A note on the gaps between zeros of Epstein's zeta-functions on the critical line

It is proved that Epstein's zeta-function $ζ_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + iγ$ with $ T \leq γ\leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive numbers $T$. This is an improvement of the result by M. Jutila and K. Srinivas (Bull. London Math. Soc. 37 (2005) 45--53).

math.NT↗
arXiv2017

Non-Wieferich primes in number fields and ABC conjecture

Let $K/\mathbb{Q}$ be an algebraic number field of class number one and $\mathcal{O}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $\mathcal{O}_K$ under the assumption of the \textit{abc} conjecture for number fields.

math.NT↗
arXiv2011

Divisibility of class numbers of imaginary quadratic function fields by a fixed odd number

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field $\mathbb{F}_{q}(x)$ whose class groups have elements of a fixed odd order. More precisely, for $q$, a power of an odd prime, and $g$ a fixed odd positive integer $\ge 3$, we show that for every $ε>0$, there are $\gg q^{L(1/2+\frac{3}{2(g+1)}-ε)}$ polynomials $f \in \mathbb{F}_{q}[x]$ with $°f=L$, for which the class group of the quadratic extension $\mathbb{F}_{q}(x, \sqrt{f})$ has an element of order $g$. This sharpens the previous lower bound $q^{L(1/2+\frac{1}{g})}$ of Ram Murty. Our result is a function field analogue to a similar result of Soundararajan for number fields.

math.NT↗