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Ritabrata Munshi

Publications and source records attributed to Ritabrata Munshi.

At least 19 recordsLinked to original sources
arXiv2025

Sub-Weyl bound for $GL(2)$ via trivial delta

For a $SL(2,\mathbb{Z})$ form $f$, we obtain the sub-Weyl bound \begin{equation*} L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-δ+\varepsilon}, \end{equation*} where $δ=1/174$, thereby crossing the Weyl barrier for the first time beyond $GL(1)$. The proof uses a refinement of the `trivial' delta method.

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arXiv2023

Sub-convexity bound for $GL(3) \times GL(2)$ $L$-functions: Hybrid level aspect

Let $F$ be a $G L(3)$ Hecke-Maass cusp form of prime level $P_1$ and let $f$ be a $G L(2)$ Hecke-Maass cuspform of prime level $P_2$. In this article, we will prove a subconvex bound for the $G L(3) \times G L(2)$ Rankin-Selberg $L$-function $L(s,F\times f)$ in the level aspect for certain ranges of the parameters $P_1$ and $P_2$.

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arXiv2023

Subconvexity for Symmetric Square L-functions off-centre

Let $p$ be a prime. Let $f$ be a holomorphic modular form of level $p$ with trivial nebentypus. We prove the bound $L\left(\text{sym}^2f, \frac{1}{2} + it\right) \ll_{f,ε} p^{1/2+ε}t^{3/4-1/12 + ε}$. This bound is subconvex in the $t$-aspect and almost convex in the level aspect simultaneously.

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arXiv2021

Beyond the Weyl barrier for $\mathrm{GL}(2)$ exponential sums

In this paper, we use the Bessel $δ$-method, along with new variants of the van der Corput method in two dimensions, to prove non-trivial bounds for $\mathrm{GL}(2)$ exponential sums beyond the Weyl barrier. More explicitly, for sums of $\mathrm{GL}(2)$ Fourier coefficients twisted by $e(f(n))$, with length $N$ and phase $f(n)=N^β \log n / 2π$ or $a n^β$, non-trivial bounds are established for $ β< 1.63651... $, which is beyond the Weyl barrier at $β= 3/2$.

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arXiv2018

Subconvexity for $GL(3)\times GL(2)$ $L$-functions in $t$-aspect

Let $π$ be a Hecke-Maass cusp form for $SL(3,\mathbb Z)$ and $f$ be a holomorphic (or Maass) Hecke form for $SL(2,\mathbb{Z})$. In this paper we prove the following subconvex bound $$ L\left(\tfrac{1}{2}+it,π\times f\right)\ll_{π,f,\varepsilon} (1+|t|)^{\frac{3}{2}-\frac{1}{42}+\varepsilon}. $$

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arXiv2017

Subconvexity for symmetric square $L$-functions

Let $f$ be a holomorphic modular form of prime level $p$ and trivial nebentypus. We show that there exists a computable $δ>0$, such that $$ L\left(\tfrac{1}{2},\mathrm{Sym}^2 f\right)\ll p^{\tfrac{1}{2}-δ}, $$ with the implied constant depending only on $δ$ and the weight of $f$.

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arXiv2017

W\lowercase{eyl} \lowercase {bound for $p$-power twist of} $GL(2)$ L-\lowercase{functions }

Let $f$ be a cuspidal eigenform (holomorphic or Maass) on the full modular group $SL(2, \mathbb{Z})$ . Let $χ$ be a primitive character of modulus $P$. We shall prove the following results: 1. Suppose $P = p^r$, where $p$ is a prime and $r\equiv 0 (\textrm{mod} \ 3)$. Then we have \[ L\left( f \otimes χ, \frac{1}{2}\right) \ll_{f, ε} P^{1/3 +ε}, \] where $ε> 0$ is any positive real number. 2. Suppose $χ$ factorizes as $χ= χ_1 χ_2$, where $ χ_i$'s are primitive character modulo $P_i$, where $P_i$ are primes, $P^{1/2 -ε} \ll P_i \ll P^{1/2 + ε}$ for $i=1,2$ and $P=P_1 P_2$. We have the Burgess bound \[ L\left( f \otimes χ, \frac{1}{2}\right) \ll_{f, ε} P^{3/8 +ε}, \] where $ε> 0$ is any positive real number.

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arXiv2016

Twists of $GL(3)$ $L$-functions

Let $π$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form, and let $χ$ be a primitive Dirichlet character modulo $M$, which we assume to be prime. In this note we revisit the subconvexity problem addressed in `The circle method and bounds for $L$-functions IV' and establish the following unconditional bound \begin{align*} L\left(\tfrac{1}{2},π\otimesχ\right)\ll M^{3/4-1/308+\varepsilon}. \end{align*}

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arXiv2015

A case of simultaneous non-vanishing

We show that for $k>1000$ an even number and a sufficiently large prime $q$, there exists a newform $f$ of weight $k$ and level $q$ such that $$ L(1/2,f)L(1/2,\text{Sym}^2 f)\neq 0. $$

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arXiv2014

Character sums of composite moduli and hybrid subconvexity

Let $M=M_1 M_2 M_3$ be the product of three distinct primes and let $χ=χ_1 χ_2 χ_3$ be a Dirichlet character of modulus $M$ such that each $χ_i$ is a primitive character modulo $M_i$ for $i=1,2,3$. In this paper, we provide a $δ$-symbol method for obtaining non-trivial cancellation in smooth character sums of the form $\sum_{n=1}^\infty χ(n) W(n/N)$, with $N$ roughly of size $\sqrt M$ and $W$ a smooth compactly supported weight function on $(0, \infty)$. As a corollary, we establish hybrid subconvexity bounds for the associated Dirichlet $L$-function.

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arXiv2014

Pairs of quadrics in 11 variables

For non-singular intersections of pairs of quadrics in 11 or more variables, we prove an asymptotic for the number of rational points in an expanding box.

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arXiv2014

The circle method and bounds for $L$-functions - IV: Subconvexity for twists of $GL(3)$ $L$-functions - B

Let $π$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form satisfying the Ramanujan conjecture and the Selberg-Ramanujan conjecture, and let $χ$ be a primitive Dirichlet character modulo $M$, which we assume to be prime for simplicity. We will prove the following subconvex bound $$ L\left(\tfrac{1}{2},π\otimesχ\right)\ll_{π,\varepsilon} M^{\frac{3}{4}-\frac{1}{1612}+\varepsilon}. $$

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