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Saurabh Kumar Singh

Publications and source records attributed to Saurabh Kumar Singh.

At least 19 recordsLinked to original sources
arXiv2026

Sum of the $GL(3)$ Fourier coefficients over mixed powers

Let $A(n)$ be the $(1,n)$-th Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke-Maass cusp form, denoted as $A(1,n)$ or the triple divisor function, denoted as $d_3(n)$. Let $k \geqslant3$ be an integer. In this paper, we establish an asymptotic formula for the sum \begin{equation*} \mathop{\sum}_{\substack{1 \leqslant n_1, n_2 \leqslant X^{1/2} \\ 1 \leqslant n_3 \leqslant X^{1/k}}} A(Q(n_1,n_2) + n_3^k)\mathsf{a}(n_3), \end{equation*} where $\mathsf{a}(n)$ is either von-Mangoldt function or identity function, and $Q(x,y) \in \mathbb{Z}[x,y]$ is a binary quadratic polynomial. When $A(n)=A(1,n)$, then $\mathsf{a}(n)$ can be any bounded arithmetical function.

math.NT↗
arXiv2026

Sum of the $GL(3)$ Fourier coefficients over quadratics

Let $A(n)$ denote the $(1,n)\text{-th}$ Fourier coefficient of a $SL(3, \mathbb{Z})$ Hecke eigenform or the ternary divisor function $d_3(n)$. Let $Q(x,y)$ be a symmetric positive definite quadratic form. This article establishes an asymptotic formula with a power-saving error term for the following sum \begin{equation*} \sum_{1 \leqslant m \leqslant X} \sum_{1 \leqslant n\leqslant Y} A(Q(m,n)), \end{equation*} where $X>1$ and $Y\leqslant X$.

math.NT↗
arXiv2024

Hybrid subconvexity bound for $GL(3)\times GL(2)$ $L$-functions: t and level aspect

\begin{abstract} In this article, we will get non-trivial estimates for the central values of degree six Rankin-Selberg $L$-functions $L(1/2+it, π\times f)$ associated with a ${GL(3)}$ form $π$ and a ${GL(2)} $ form $f$ using the delta symbol approach in the hybrid settings i.e. in the level of ${GL(2)}$ form and $t$-aspect. \end{abstract}

math.NT↗
arXiv2023

Shifted Convolution Sum for $GL(3) \times GL(2)$ with Weighted Average

In this paper, we will prove the non-trivial bound for the weighted average version of shifted convolution sum for $GL(3)\times GL(2)$, i.e. for any $ε>0$ and $X^{1/4+δ} \leq H \leq X$ with $δ>0$, \[ \frac{1}{H}\sum_{h=1}^\infty λ_f(h) V\left( \frac{h}{H}\right)\sum_{n=1}^\infty λ_π(1,n) λ_g (n+h) W\left( \frac{n}{X} \right)\ll X^{1-δ+ε} \] where $V,W$ are smooth compactly supported funtions, $λ_f(n), λ_g(n)$ and $λ_π(1,n)$ are the normalized n-th Fourier coefficients of $SL(2,\mathbb{Z})$ Hecke-Maass cusp forms $f,g$ and $SL(3,\mathbb{Z})$ Hecke-Maass cusp form $π$, respectively.

math.NT↗
arXiv2023

Moments of derivatives of modular $L$-functions

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}.

math.NT↗
arXiv2023

A versatile and narrow linewidth infra-red radiation source for ro-vibration state selected preparation of molecules in molecular beams

We describe the design and characterization of a versatile pulsed (5 ns, 10 Hz repetition rate) optical parametric oscillator and amplifier system capable of generating single longitudinal mode, narrow linewidth (0.01 cm$^{-1}$) radiation in the wavelength range of 680 - 870 nm and 1380 - 4650 nm. Using a combination of power-normalized photoacoustic signal and a Fizeau interferometer-based wavemeter, we are able to actively stabilize the output wavenumber to within 0.005 cm$^{-1}$ (3$σ$) over a timescale longer than 1000 seconds. We demonstrate an application of this system by performing ro-vibration state-selected preparation of CO in v = 2 state, via direct overtone excitation (v = 0 $\rightarrow 2$ at 2346 nm) and subsequent state-selected detection in an internally cold molecular beam.

physics.optics↗
arXiv2023

The curious case of CO$_2$ dissociation on Cu(110)

Dissociation of CO$_2$ on copper surfaces, a model system for understanding the elementary steps in catalytic conversion of CO$_2$ to methanol has been extensively studied in the past. It is thought to be reasonably well-understood from both experiments and theory. In contrast, our findings reported here suggest a different picture. Using molecular beam surface scattering methods, we measure the initial dissociation probabilities ($S_{\rm 0}$) of CO$_2$ on a flat, clean Cu(110) surface under ultra-high vacuum conditions. The observed $S_{\rm 0}$ ranges from $3.9\times10^{-4}$ to $1.8\times10^{-2}$ at incidence energies of 0.64 eV to 1.59 eV with a lower limit to dissociation barrier estimated to be around 2.0 eV, much larger than that understood previously. We discuss the possible reasons behind such large differences in our results and previous work. These findings are anticipated to be extremely important for obtaining a correct understanding of elementary steps in CO$_2$ dissociation on Cu surfaces.

physics.chem-ph↗
arXiv2023

Subconvex bound for $\textrm{GL(3)} \times \textrm{GL(2)}$ $L$-functions: $\textrm{GL(3)}$-spectral aspect

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 $0 < ξ<1/2$.

math.NT↗
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 $P_2$.

math.NT↗
arXiv2023

A compact and highly collimated atomic/molecular beam source

We describe the design, characterization and application of a simple, highly collimated and compact atomic/molecular beam source. This source is based on a segmented capillary design, constructed using a syringe needle. Angular width measurements and free molecular flow simulations show that the segmented structure effectively suppresses atoms travelling in off-axis directions, resulting in a narrow beam of Helium atoms having a width of 7 mrad (full width half maximum). We demonstrate an application of this source by using it for monitoring real-time changes in surface coverage on a clean Cu(110) surface exposed to oxygen, by measuring specular reflectivity of the Helium beam generated using this source.

physics.chem-ph↗
arXiv2022

A simple and low-cost setup for part per billion level frequency stabilization and characterization of red He-Ne laser

This work describes the frequency stabilization of a dual longitudinal mode, red (632.8 nm) He-Ne laser, implemented using a low-cost microcontroller and its performance characterization using a simple interferometric method. Our studies demonstrate that frequency stability up to 0.42 MHz (3$σ$, 17 hours) can be achieved using this set up. This simple and low-cost system can serve as an excellent part per billion level frequency reference for several high resolution spectroscopy based applications.

physics.ins-det↗
arXiv2021

Non-linear additive twist of Fourier coefficients of $GL(3) \times GL(2)$ and $GL(3)$ Maass forms

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 $V^{(j)}(x) \ll_{j} 1$.

math.NT↗
arXiv2018

Subconvexity bound for $GL(2)$ L-functions: \lowercase{t}-aspect

Let $f $ be a holomorphic Hecke eigenform or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L \left( \frac{1}{2} + it, f \right) \ll_{f, ε} \left( 2 + |t|\right)^{1/3 + ε}, \] for any $ε> 0.$

math.NT↗
arXiv2018

t-Aspect Subconvexity Bound for $GL(2)$ L-functions

Let $f $ be a holomorphic Hecke eigenforms or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L \left( \frac{1}{2} + it \right) \ll_{f, ε} \left( 1 + |t|\right)^{1/3 + ε}, \] for any $ε> 0.$

math.NT↗