arXiv subjects
Publications and source records attributed to Kummari Mallesham.
Let $f$ be an Hecke eigenform for the group $Γ_{0}(q)$ and $χ_{d}$ be a primitive quadratic character of conductor $|d|$. In this article, we prove an asymptotic for the second moment of the derivative of $L(s, f \otimes χ_{8d})$ at the central point $1/2$, which was previously known under GRH by Petrow \cite{petrow}.
Let $ϕ$ be a Hecke-Maass cusp form for $\mathrm{SL(3, \mathbb{Z})}$ with Langlands parameters $({\bf t}_{i})_{i=1}^{3}$ and $f$ be a holomorphic or Hecke-Maass cusp form for $\mathrm{SL(2,\mathbb{Z})}$. In this article, we prove the following subconvex bound $$ L\left(ϕ\times f, 1/2\right) \ll_{f,ε} T^{ \frac{3}{2}-δ_ξ+ε},\ δ_ξ=\min\{ξ/4, \, (1-2ξ)/4 \}, $$ for the central value $ L\left(ϕ\times f, 1/2\right) $ in the $\mathrm{GL(3)}$-spectral aspect, where $({\bf t}_{i})_{i=1}^{3}$ satisfies $$|{\bf t}_{3} - {\bf t}_{2}| \asymp T^{1-ξ}, \quad \, {\bf t}_{i} \asymp T, \quad \, \, i=1,\,2,\,3,$$ with $ξ$ a real number such that $0 < ξ<1/2$.
Let $$S(N) = \sum_{n \sim N}^{\text{smooth}} \, λ_{f}(n) \, χ(n),$$ where $λ_{f}(n)$'s are Fourier coefficients of Hecke-eigen form, and $χ$ is a primitive character of conductor $p^{r}$. In this article we prove a sub-Weyl strength bounds for $S(N)$. Indeed, we obtain $$S(N) \ll \, N^{\frac{5}{9}} \ p^{\frac{13r}{45}},$$ provided that $ p^{13r/20} \leq N \leq p^{4r/5}$. Note that the above bound for $S(N)$ is non-trivial if $N\geq \left(p^{r}\right)^{\frac{2}{3}-\frac{1}{60}}$.
In this article, we will prove subconvex bounds for $GL(3) \times GL(2)$ $L$-functions in the depth aspect.
Let $λ_π(m,n)$ be the Fourier coefficients of a Hecke-Maass cusp form $π$ for $SL(3,\mathbb{Z})$ and $λ_{f}(n)$ be the Fourier coefficients of Hecke-eigen form $f$ for $SL(2,\mathbb{Z})$. The aim of this article is to get a non-trivial bound on the sum which is non-linear additive twist of the coefficients $λ_π(m,n)$ and $λ_{f}(n)$. More precisely, for any $0 < β< 1$ and $ε>0$, we have $$\sum_{n=1}^{\infty} λ_π(r,n) \, e\left(αn^β\right) V\left(\frac{n}{X}\right) \ll_{π,ε} α\sqrtβr^{\frac{7}{6}}X^{\frac{3}{4}+\frac{9β}{28}+ ε}.$$ and $$\sum_{n=1}^{\infty} λ_π(r,n) \, λ_{f}(n) \, e\left(αn^β\right) V\left(\frac{n}{X}\right) \ll_{π, f,ε} (αβ)^{\frac{3}{2}} rX^{\frac{3}{4}+\frac{29β}{44}+ε},$$ where $V(x)$ is a smooth function supported in $[1,2]$ and satisfying $V^{(j)}(x) \ll_{j} 1$.
We study simultaneous non-vanishing of $L(\tfrac{1}{2},\di)$ and $L(\tfrac{1}{2},g\otimes \di)$, when $\di$ runs over an orthogonal basis of the space of Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ and $g$ is a fixed $SL(2,\mathbb{Z})$ Hecke cusp form of weight $k\equiv 0 \pmod 4$.
In this article, we obtain an upper bound for the number of integral points on the del Pezzo surfaces of degree two.
In this article, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an improvement over the bound given by Henryk Iwaniec and Ritabrata Munshi in \cite{H-R}.
When the sequences of squares of primes is coloured with $K$ colours, where $K \geq 1$ is an integer, let $s(K)$ be the smallest integer such that each sufficiently large integer can be written as a sum of no more than $s(K)$ squares of primes, all of the same colour. We show that $s(K) \ll K \exp\left(\frac{(3\log 2 + {\rm o}(1))\log K}{\log \log K}\right)$ for $K \geq 2$. This improves on $s(K) \ll_ε K^{2 +ε}$, which is the best available upper bound for $s(K)$.
We obtain an upper bound for the number of pairs $ (a,b) \in {A\times B} $ such that $ a+b $ is a prime number, where $ A, B \subseteq \{1,...,N \}$ with $|A||B| \, \gg \frac{N^2}{(\log {N})^2}$, $\, N \geq 1$ an integer. This improves on a bound given by Balog, Rivat and Sárközy.