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Rajat Gupta

Publications and source records attributed to Rajat Gupta.

33 records · Page 2Linked to original sources
arXiv2021

Koshliakov zeta functions I: Modular Relations

We examine an unstudied manuscript of N.~S.~Koshliakov over $150$ pages long and containing the theory of two interesting generalizations $ζ_p(s)$ and $η_p(s)$ of the Riemann zeta function $ζ(s)$, which we call \emph{Koshliakov zeta functions}. His theory has its genesis in a problem in the analytical theory of heat distribution which was analyzed by him. In this paper, we further build upon his theory and obtain two new modular relations in the setting of Koshliakov zeta functions, each of which gives an infinite family of identities, one for each $p\in\mathbb{R^{+}}$. The first one is a generalization of Ramanujan's famous formula for $ζ(2m+1)$ and the second is an elegant extension of a modular relation on page $220$ of Ramanujan's Lost Notebook. Several interesting corollaries and applications of these modular relations are obtained including a new representation for $ζ(4m+3)$.

math.NT↗
arXiv2021

Extended higher Herglotz functions I. Functional equations

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function $F(x)$ which is now known as the \emph{Herglotz function}. As demonstrated by Zagier, and very recently by Radchenko and Zagier, $F(x)$ satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study $\mathscr{F}_{k,N}(x)$, an extension of the Herglotz function which also subsumes \emph{higher Herglotz function} of Vlasenko and Zagier. We call it the \emph{extended higher Herglotz function}. It is intimately connected with a certain generalized Lambert series. We derive two different kinds of functional equations satisfied by $\mathscr{F}_{k,N}(x)$. Radchenko and Zagier gave a beautiful relation between the integral $\displaystyle\int_{0}^{1}\frac{\log(1+t^x)}{1+t}\, dt$ and $F(x)$ and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between $\mathscr{F}_{k,N}(x)$ and a generalization of the above integral involving polylogarithm. The asymptotic expansions of $\mathscr{F}_{k, N}(x)$ and some generalized Lambert series are also obtained along with other supplementary results.

math.NT↗
arXiv2021

On a certain divisor function in Number fields

The main aim of this paper is to study an analogue of the generalized divisor function in a number field $\mathbb{K}$, namely, $σ_{\mathbb{K},α}(n)$. The Dirichlet series associated to this function is $ζ_{\mathbb{K}}(s)ζ_{\mathbb{K}}(s-α)$. We give an expression for the Riesz sum associated to $σ_{\mathbb{K},α}(n),$ and also extend the validity of this formula by using convergence theorems. As a special case, when $\mathbb{K}=\mathbb{Q}$, the Riesz sum formula for the generalized divisor function is obtained, which, in turn, for $α=0$, gives the Vorono\"ı summation formula associated to the divisor counting function $d(n)$. We also obtain a big $O$-estimate for the Riesz sum associated to $σ_{\mathbb{K},α}(n)$.

math.NT↗
arXiv2021

An Innovative Security Strategy using Reactive Web Application Honeypot

Nowadays, web applications have become most prevalent in the industry, and the critical data of most organizations stored using web apps. Hence, web applications a much bigger target for diverse cyber-attacks, which varies from database injections-SQL injection, PHP object injection, template injection, XML external entity injection, unsanitized input attacks- Cross-Site Scripting(XSS), and many more. As mitigation for them, among many proposed solutions, web application honeypots are a much sophisticated and powerful protection mechanism. In this paper, we propose a low interaction, adaptive, and dynamic web application honeypot that imitates the vulnerabilities through HTTP events. The honeypot is built with SNARE and TANNER; SNARE creates the attack surface and sends the requests to TANNER, which evaluates them and decides how SNARE should respond to the requests. TANNER is an analysis and classification tool, which analyzes and evaluates HTTP requests served by SNARE and to compose the response, it is powered by emulators, which are engines used for the emulation of vulnerabilities.

cs.CR↗
arXiv2021

Generalizations of the Andrews-Yee identities associated with the mock theta functions $ω(q)$ and $ν(q)$

George Andrews and Ae Ja Yee recently established beautiful results involving bivariate generalizations of the third order mock theta functions $ω(q)$ and $ν(q)$, thereby extending their earlier results with the second author. Generalizing the Andrews-Yee identities for trivariate generalizations of these mock theta functions remained a mystery, as pointed out by Li and Yang in their recent work. We partially solve this problem and generalize these identities. Several new as well as well-known results are derived. For example, one of our two main theorems gives, as a corollary, a special case of Soon-Yi Kang's three-variable reciprocity theorem. A relation between a new restricted overpartition function $p^{*}(n)$ and a weighted partition function $p_*(n)$ is obtained from one of the special cases of our second theorem.

math.CO↗
arXiv2020

On some $q$-series identities related to a generalized divisor function and their implications

In this article, a $q$-series examined by Kluyver and Uchimura is generalized. This allows us to find generalization of the identities in the random acyclic digraph studied by Simon, Crippa, and Collenberg in 1993. As one of the corollaries of our main theorem, we get results of Dilcher and Andrews, Crippa, and Simon. This main theorem involves a surprising new generalization of the divisor function $σ_s(n)$, which we denote by $σ_{s,z}(n)$. Analytic properties of $σ_{s,z}(n)$ are also studied. As a special case of one of our theorem we obtain result from a recent paper of Bringmann and Jennings-Shaffer.

math.NT↗
arXiv2020

AIM 2020 Challenge on Real Image Super-Resolution: Methods and Results

This paper introduces the real image Super-Resolution (SR) challenge that was part of the Advances in Image Manipulation (AIM) workshop, held in conjunction with ECCV 2020. This challenge involves three tracks to super-resolve an input image for $\times$2, $\times$3 and $\times$4 scaling factors, respectively. The goal is to attract more attention to realistic image degradation for the SR task, which is much more complicated and challenging, and contributes to real-world image super-resolution applications. 452 participants were registered for three tracks in total, and 24 teams submitted their results. They gauge the state-of-the-art approaches for real image SR in terms of PSNR and SSIM.

cs.CV↗
arXiv2020

NTIRE 2020 Challenge on Real Image Denoising: Dataset, Methods and Results

This paper reviews the NTIRE 2020 challenge on real image denoising with focus on the newly introduced dataset, the proposed methods and their results. The challenge is a new version of the previous NTIRE 2019 challenge on real image denoising that was based on the SIDD benchmark. This challenge is based on a newly collected validation and testing image datasets, and hence, named SIDD+. This challenge has two tracks for quantitatively evaluating image denoising performance in (1) the Bayer-pattern rawRGB and (2) the standard RGB (sRGB) color spaces. Each track ~250 registered participants. A total of 22 teams, proposing 24 methods, competed in the final phase of the challenge. The proposed methods by the participating teams represent the current state-of-the-art performance in image denoising targeting real noisy images. The newly collected SIDD+ datasets are publicly available at: https://bit.ly/siddplus_data.

cs.CV↗
arXiv2020

On Sum-Of-Tails Identities

In this article, a finite analogue of the generalized sum-of-tails identity of Andrews and Freitas is obtained. We derive several interesting results as special cases of this analogue, in particular, a recent identity of Dixit, Eyyyunni, Maji and Sood. We derive a new extension of Abel's lemma with the help of which we obtain a one-parameter generalization of a sum-of-tails identity of Andrews, Garvan and Liang, an identity of Ramanujan as well as two new results - one for Ramanujan's function $σ(q)$ and another for the function recently introduced by Andrews and Ballantine. Later we introduce a new generalization $\mathrm{FFW}_{c}(n)$ of a function of Fokkink, Fokkink and Wang and derive an identity for its generating function. This gives, as a special case, a recent representation for the generating function of $\mathrm{spt}(n)$ given by Andrews, Garvan and Liang. We also obtain some weighted partition identities along with new representations for two of Ramanujan's third order mock theta functions through combinatorial techniques.

math.CO↗
arXiv2019

Delog: A Privacy Preserving Log Filtering Framework for Online Compute Platforms

In many software applications, logs serve as the only interface between the application and the developer. However, navigating through the logs of long-running applications is often challenging. Logs from previously successful application runs can be leveraged to automatically identify errors and provide users with only the logs that are relevant to the debugging process. We describe a privacy preserving framework which can be employed by Platform as a Service (PaaS) providers to utilize the user logs generated on the platform while protecting the potentially sensitive logged data. Further, in order to accurately and scalably parse log lines, we present a distributed log parsing algorithm which leverages Locality Sensitive Hashing (LSH). We outperform the state-of-the-art on multiple datasets. We further demonstrate the scalability of Delog on publicly available Thunderbird log dataset with close to 27,000 unique patterns and 211 million lines.

cs.IR↗
arXiv2019

A Ramanujan-type formula for $ζ^{2}(2m+1)$ and its generalizations

A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for $ζ(2m+1)$. The formula for $ζ^{2}(2m+1)$ is then generalized in two different directions, one, by considering the generalized divisor function $σ_z(n)$, and the other, by studying a more general analogue of the aforementioned Eisenstein series, consisting of one more parameter $N$. A number of important special cases are derived from the first generalization. For example, we obtain a series representation for $ζ(1+ω)ζ(-1-ω)$, where $ω$ is a non-trivial zero of $ζ(z)$. We also evaluate a series involving the modified Bessel function of the second kind in the form of a rational linear combination of $ζ(4k-1)$ and $ζ(4k+1)$ for $k\in\mathbb{N}$.

math.NT↗
arXiv2018

Generalized Lambert series, Raabe's integral and a two-parameter generalization of Ramanujan's formula for $ζ(2m+1)$

A comprehensive study of the generalized Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}\exp{(-an^{N}x)}}{1-\exp{(-n^{N}x)}}, 0 0$, $N\in\mathbb{N}$ and $h\in\mathbb{Z}$, is undertaken. Two of the general transformations of this series that we obtain here lead to two-parameter generalizations of Ramanujan's famous formula for $ζ(2m+1)$, $m>0$ and the transformation formula for $\logη(z)$. Numerous important special cases of our transformations are derived. An identity relating $ζ(2N+1), ζ(4N+1),\cdots, ζ(2Nm+1)$ is obtained for $N$ odd and $m\in\mathbb{N}$. Certain transcendence results of Zudilin- and Rivoal-type are obtained for odd zeta values and generalized Lambert series. A criterion for transcendence of $ζ(2m+1)$ and a Zudilin-type result on irrationality of Euler's constant $γ$ are also given. New results analogous to those of Ramanujan and Klusch for $N$ even, and a transcendence result involving $ζ\left(2m+1-\frac{1}{N}\right)$, are obtained.

math.NT↗
arXiv2017

Underlying Event Measurements at CMS

Measurements of Underlying Event activity using proton-proton collision data collected by the CMS detector will be presented. To check the energy dependence of the underlying event activity, results are compared with previous measurements from different experiments at different centre-of-mass energies.

hep-ex↗
arXiv2017

An Improved Fatigue Detection System Based on Behavioral Characteristics of Driver

In recent years, road accidents have increased significantly. One of the major reasons for these accidents, as reported is driver fatigue. Due to continuous and longtime driving, the driver gets exhausted and drowsy which may lead to an accident. Therefore, there is a need for a system to measure the fatigue level of driver and alert him when he/she feels drowsy to avoid accidents. Thus, we propose a system which comprises of a camera installed on the car dashboard. The camera detect the driver's face and observe the alteration in its facial features and uses these features to observe the fatigue level. Facial features include eyes and mouth. Principle Component Analysis is thus implemented to reduce the features while minimizing the amount of information lost. The parameters thus obtained are processed through Support Vector Classifier for classifying the fatigue level. After that classifier output is sent to the alert unit.

cs.CV↗