arXiv subjects

Subham De

Publications and source records attributed to Subham De.

10 recordsLinked to original sources

arXiv2024

On Proving Ramanujan's Inequality using a Sharper Bound for the Prime Counting Function $\pi(x)$

This article provides a proof that the Ramanujan's Inequality given by, $$\pi(x)^2 < \frac{e x}{\log x} \pi\Big(\frac{x}{e}\Big)$$ holds unconditionally for every $x\geq \exp(43.5102147)$. In case for an alternate proof of the result stated above, we shall exploit certain estimates involving the Chebyshev Theta Function, $\vartheta(x)$ in order to derive appropriate bounds for $\pi(x)$, which'll lead us to a much improved condition for the inequality proposed by Ramanujan to satisfy unconditionally.

math.NT↗

arXiv2024

Inequalities involving Higher Degree Polynomial Functions in $\pi(x)$

The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $\pi(x)$ having a general expression of the form, \begin{align*} P(\pi(x)) - \frac{e x}{\log x} Q(\pi(x/e)) + R(x) \end{align*} $P$, $Q$ and $R$ are arbitrarily chosen polynomials and $\pi(x)$ denotes the \textit{Prime Counting Function}. The proofs require specific order estimates involving $\pi(x)$ and the \textit{Second Chebyshev Function} $\psi(x)$, as well as the famous \textit{Prime Number Theorem} in addition to certain meromorphic properties of the \textit{Riemann Zeta Function} $\zeta(s)$ and results regarding its non-trivial zeros. A few generalizations of these concepts have also been discussed in detail towards the later stages of the paper, along with citing some important applications.

math.GM↗

arXiv2024

On proving an Inequality of Ramanujan using Explicit Order Estimates for the Mertens Function

This research article provides an unconditional proof of an inequality proposed by Srinivasa Ramanujan involving the Prime Counting Function $\pi(x)$, \begin{align*} (\pi(x))^{2}<\frac{ex}{\log x}\pi\left(\frac{x}{e}\right) \end{align*} for every real $x\geq \exp(547)$, using specific order estimates for the Mertens Function, $M(x)$. The proof primarily hinges upon investigating the underlying relation between $M(x)$ and the Second Chebyshev Function, $\psi(x)$, in addition to applying the meromorphic properties of the Riemann Zeta Function, $\zeta(s)$ with an intention of deriving an improved approximation for $\pi(x)$.

math.GM↗

arXiv2023

A Detailed Study of Kirchhoff-type Critical Elliptic Equations and $p$-Sub-Laplacian Operators within the Heisenberg Group $\mathcal{H}_{n}$ Framework

This article presents a comprehensive study of \textit{Kirchhoff-type Critical Elliptic Equations} involving $p$-sub-Laplacian Operators on the \textit{Heisenberg Group} $\mathcal{H}_{n}$. It delves into the mathematical framework of Heisenberg Group, and explores their Spectral Properties. A significant focus is on the existence and multiplicity of solutions under various conditions, leveraging concepts like the \textit{Mountain Pass Theorem}. This work not only contributes to the theoretical understanding of such groups but also has implications in fields like Quantum Mechanics and Geometric Group Theory.

math.GM↗

arXiv2023

On the Order Estimates for Specific Functions of $\zeta(s)$ and its Contribution towards the Analytic Proof of The Prime Number Theorem

This article provides a proof of the famous \textit{Prime Number Theorem} by establishing an analogous statement of the same in terms of the second \textit{Chebyshev Function} $\psi(x)$. We shall be extensively using complex analytic techniques in addition to certain meromorphic properties of the \textit{Reimann Zeta Function} $\zeta(s)$ and its \textit{Analytic Continuation Property} a priori using Riemann's Functional Equation in order to establish our desired result.

math.GM↗

arXiv2023

On the proof of the Prime Number Theorem using Order Estimates for the Chebyshev Theta Function

In this paper, we shall study the stellar work of Norwegian mathematician Selberg and Hungarian mathematician Erd\H{o}s in providing an Elementary proof of the well-known \textit{Prime Number Theorem}. In addition to introducing ourselves to the notion of \textit{Arithmetic Functions}, we shall primarily focus our research on obtaining suitable estimates for the \textit{Chebyshev Theta Function} $\vartheta(x)$. Furthermore, we'll try to infer about the asymptotic properties of another function $\rho(x)$, which shall be needed later on in establishing an equivalent statement of our main result. All the mathematical terminologies pertinent to the proof have been discussed in the earlier sections of the text.

math.HO↗

arXiv2023

A Brief Introduction To Splitting Of Primes Over Number Fields

The study of \textit{Dedekind Zeta Functions} over a number field extension uses different aspects of both \textit{Algebraic} and \textit{Analytic Number Theory}. In this paper, we shall learn about the structure and different analytic aspects of such functions, namely the domain of its convregence and analyticity at different points of $\mathbb{C}$ when the function is defined over any finite field extension $K$ over $\mathbb{Q}$ . Moreover, given any two Number Fields $L$ and $K$ over $\mathbb{Q}$ with $L$ being Normal over $K$, our intention is to classify and study the primes in $K$ which split completely in $L$. Also, we shall explore some special cases related to this result.

math.HO↗

arXiv2023

Meromorphic Continuation Of Global Zeta Function For Number Fields

In the paper, we shall establish the existence of a meromorphic continuation of the Global Zeta Function $\zeta(f,\chi)$ of a Global Number Field $K$ and also deduce the functional equation for the same, using different properties of the id\`ele class group $\mathcal{C}_K^1$ of a global field $K$ extensively defined using basic notions of Ad\`eles ($\mathbb{A}_{K}$) and Id\`eles ($\mathbb{I}_{K}$) of $K$, and also evaluating Fourier Transforms of functions $f$ on the space $\mathcal{S}(\mathbb{A}_{K})$ of Ad\`elic Schwartz-Bruhat Functions. A brief overview of most of the concepts required to prove our desired result have been provided to the readers in the earlier sections of the text.

math.HO↗

arXiv2023

Packet Reception Probability: Packets That You Can't Decode Can Help Keep You Safe

This paper provides a robust, scalable Bluetooth Low-Energy (BLE) based indoor localization solution using commodity hardware. While WiFi-based indoor localization has been widely studied, BLE has emerged a key technology for contact-tracing in the current pandemic. To accurately estimate distance using BLE on commercial devices, systems today rely on Receiver Signal Strength Indicator(RSSI) which suffers from sampling bias and multipath effects. We propose a new metric: Packet Reception Probability (PRP) that builds on a counter-intuitive idea that we can exploit packet loss to estimate distance. We localize using a Bayesian-PRP formulation that also incorporates an explicit model of the multipath. To make deployment easy, we do not require any hardware, firmware, or driver-level changes to off-the-shelf devices, and require minimal training. PRP can achieve meter level accuracy with just 6 devices with known locations and 12 training locations. We show that fusing PRP with RSSI is beneficial at short distances < 2m. Beyond 2m, fusion is worse than PRP, as RSSI becomes effectively de-correlated with distance. Robust location accuracy at all distances and ease of deployment with PRP can help enable wide range indoor localization solutions using BLE.

cs.NI↗

arXiv2017

Finding by Counting: A Probabilistic Packet Count Model for Indoor Localization in BLE Environments

We propose a probabilistic packet reception model for Bluetooth Low Energy (BLE) packets in indoor spaces and we validate the model by using it for indoor localization. We expect indoor localization to play an important role in indoor public spaces in the future. We model the probability of reception of a packet as a generalized quadratic function of distance, beacon power and advertising frequency. Then, we use a Bayesian formulation to determine the coefficients of the packet loss model using empirical observations from our testbed. We develop a new sequential Monte-Carlo algorithm that uses our packet count model. The algorithm is general enough to accommodate different spatial configurations. We have good indoor localization experiments: our approach has an average error of ~1.2m, 53% lower than the baseline range-free Monte-Carlo localization algorithm.

cs.NI↗