arXiv subjects

Rajat Gupta

Publications and source records attributed to Rajat Gupta.

At least 19 recordsLinked to original sources

arXiv2026

Chopping and distilling variational autoencoders for real-time anomaly detection in high energy physics

Anomaly detection (AD) has recently emerged as an exciting alternative to conventional search strategies in high energy physics using artificial intelligence (AI) and machine learning (ML). The integration of these techniques into trigger systems is even more recent, but represents a crucial step in expanding the coverage of LHC triggers. In this paper, we explore the direct comparison, as well as combination, of two compression techniques for variational autoencoder (VAE) AD trigger algorithms: utilizing only latent-space derived variables and therefore requiring only half of the VAE that we call "chopping" and applying knowledge distillation (KD) to distill the VAE into a student architecture that we call "distillation." We demonstrate the feasibility of deploying such techniques on an FPGA within the resource and latency constraints of an LHC trigger environment and further find that a combination of the two leads to the smallest models that maintain, and in some cases, improve, performance with respect to the original VAE architecture.

physics.ins-det↗

arXiv2026

Ramanujan's and Lim's Identities and Harmonic Maass--Jacobi Forms

We study an extension of Ramanujan's identities for odd zeta values by Lim and introduce Jacobi analogues of classical Eichler integrals of Eisenstein series. In negative weight we construct explicit completions and embed these objects into a modular framework by showing that they are (singular) harmonic Maass--Jacobi forms. We further describe their non-holomorphic parts in terms of Eichler integrals, establish Ramanujan-type inversion formulas, and study their behavior under the Maass raising and lowering operators and at torsion points.

math.NT↗

arXiv2026

A divisor function of Wigert and higher degree forms

Let $k\in\mathbb{N}$. Wigert's divisor function $d^{\left(\frac{1}{k}\right)}(j)$ counts the number of representations of $j$ of the form $m^k+mn$ with $m\geq1 , n\geq0$. Let $\mathcal{F}_k(s)$ denote the Dirichlet series of $d^{\left(\frac{1}{k}\right)}(j)$. While $\mathcal{F}_2(s)$ is essentially a well-known special case of the Euler-Zagier double zeta function, and hence well-studied, very little is known about $\mathcal{F}_k(s)$ for $k>2$. We offer three new representations for $\mathcal{F}_k(s)$ for $k\geq2$, one of which is an analogue of the Chowla-Selberg formula as well as of a formula of Atkinson. The meromorphicity of $\mathcal{F}_k(s)$ is also discussed. The special value $\mathcal{F}_3\left(\frac{3}{2}\right)$ is expressed in terms of an infinite series of Bessel functions and a generalized divisor function.

math.NT↗

arXiv2026

Memristive tabular variational autoencoder for compression of analog data in high energy physics

We present an implementation of edge AI to compress data on an in-memory analog content-addressable memory (ACAM) device. A variational autoencoder is trained on a simulated sample of energy measurements from incident high-energy electrons on a generic three-layer scintillator-based calorimeter. The encoding part is distilled into tabular format by regressing the latent space variables using decision trees, which is then programmed on a memristor-based ACAM. In real-time, the ACAM compresses 48 continuously valued incoming energies measured by the calorimeter sensors into the latent space, achieving a compression factor of 12x, which is transmitted off-detector for decompression. The performance result of the ACAM, obtained using the Structural Simulation Toolkit, the SST open source framework, gives a latency value of 24 ns and a throughput of 330M compressions per second, i.e., 3 ns between successive inputs, and an average energy consumption of 4.1 nJ per compression.

physics.ins-det↗

arXiv2025

Summation formulas for Hurwitz class numbers and other mock modular coefficients

We prove a formula for weighted sums of the first $n$ coefficients of mock modular forms of moderate growth and apply it to Hurwitz class numbers and coefficients of negative half integral weight Eisenstein series, which take the form of certain quadratic Dirichlet $L$-values. Our formula is a mock modular version of a Bessel-sum identity proved by Chandrasekharan and Narasimhan for Dirichlet series satisfying a functional equation. Our proof utilizes $L$-functions for mock modular Eisenstein series defined by Shankadhar and Singh.

math.NT↗

arXiv2025

Non-standard quaternary representations and the Fibonacci numbers

Let $f_4(n)$ be the number of hyperquaternary representations of $n$ and $b_4(n)$ be the number of balanced quaternary representations of $n$. We show that there is no integer $k$ such that $f_4(n+k)=b_4(n)$ for all $n\ge -k$, in contrast to the binary case. Nevertheless, there do exist integers $k$ such that $f_4(n+k)=b_4(n)$ for arbitrarily large intervals of $n$. We generalize these results to any even base $d$. We also study the rate of growth of $b_4(n)$ and show that maximal values of this function correspond to certain Fibonacci numbers.

math.NT↗

arXiv2024

A segment of Euler product associated to a certain Dirichlet series

In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function $\sigma_{\alpha}(n):=\sum_{d|n}d^{\alpha}$. We obtain an exact identity relating the Dirichlet series $\zeta(s)\zeta(s-\alpha)$ and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip.

math.NT↗

arXiv2024

Partitions in which every term but the smallest one is consecutive

In this article, we introduce the notion of almost consecutive partitions. A partition is almost consecutive if every term is consecutive, with the possible exception of the smallest one. We find formulas relating to the smallest parts of consecutive and almost consecutive partitions. We also find an alternate combinatorial interpretation of the number of almost consecutive partitions of a given integer $n$ and an asymptotic formula for this quantity.

math.CO↗

arXiv2024

On the $k$th smallest part of a partition into distinct parts

A classic theorem of Uchimura states that the difference between the sum of the smallest parts of the partitions of $n$ into an odd number of distinct parts and the corresponding sum for an even number of distinct parts is equal to the number of divisors of $n$. In this article, we initiate the study of the $k$th smallest part of a partition $π$ into distinct parts of any integer $n$, namely $s_k(π)$. Using $s_k(π)$, we generalize the above result for the $k$th smallest parts of partitions for any positive integer $k$ and show its connection with divisor functions for general $k$ and derive interesting special cases. We also study weighted partitions involving $s_k(π)$ with another parameter $z$, which helps us obtain several new combinatorial and analytical results. Finally, we prove sum-of-tails identities associated with the weighted partition function involving $s_k(π)$.

math.NT↗

arXiv2023

Non-standard binary representations and the Stern sequence

We show that the number of short binary signed-digit representations of an integer $n$ is equal to the $n$-th term in the Stern sequence. Various proofs are provided, including direct, bijective, and generating function proofs. We also show that this result can be derived from recent work of Monroe on binary signed-digit representations of a fixed length.

math.CO↗

arXiv2023

A note on odd zeta values over any number field and Extended Eisenstein series

In this article, we have studied transformation formulas of zeta function at odd integers over an arbitrary number field which in turn generalizes Ramanujan's identity for the Riemann zeta function. The above transformation leads to a new number field extension of Eisenstein series, which satisfies the transformation $z \mapsto -1/z$ like an integral weight modular form over SL$_2(\Z)$. The results provide number of important applications, which are important in studying the behaviour of odd zeta values as well as Lambert series in an arbitrary number field.

math.NT↗

arXiv2023

Modified Bessel Functions in Analytic Number Theory

The modified Bessel functions $K_ν(z)$, or, for brevity, K-Bessel functions, arise at key places in analytic number theory. In particular, they appear in beautiful arithmetic identities. A survey of these arithmetical identities and their appearances in number theory is provided.

math.NT↗

arXiv2022

Two General Series Identities Involving Modified Bessel Functions and a Class of Arithmetical Functions

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{λ_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{μ_n^{s}},$$ satisfying a familiar functional equation involving the gamma function $Γ(s)$. Two general identities are established. The first involves the modified Bessel function $K_μ(z)$, and can be thought of as a 'modular' or 'theta' relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_μ(z)$, the Bessel functions of imaginary argument $I_μ(z)$, and ordinary hypergeometric functions ${_2F_1}(a,b;c;z)$. Although certain special cases appear in the literature, the general identities are new. The arithmetical functions appearing in the identities include Ramanujan's arithmetical function $τ(n)$; the number of representations of $n$ as a sum of $k$ squares $r_k(n)$; and primitive Dirichlet characters $χ(n)$.

math.NT↗

arXiv2022

Extended Higher Herglotz function \textup{II}

Very recently, Radchenko and Zagier revived the theory of Herglotz functions. The main goal of the article is to show that one of the formulas on page 220 of Ramanujan's Lost Notebook actually lives in the realms of this theory. As a consequence of our general theorem, we derive an interesting identity analogous to Ramanujan's formula for $ζ(2m+1)$. We also introduce a character analogue of the Herglotz function and initiate its theory by obtaining an elegant functional equation governed by it.

math.NT↗

arXiv2021

SOK: On the Analysis of Web Browser Security

Web browsers are integral parts of everyone's daily life. They are commonly used for security-critical and privacy sensitive tasks, like banking transactions and checking medical records. Unfortunately, modern web browsers are too complex to be bug free (e.g., 25 million lines of code in Chrome), and their role as an interface to the cyberspace makes them an attractive target for attacks. Accordingly, web browsers naturally become an arena for demonstrating advanced exploitation techniques by attackers and state-of-the-art defenses by browser vendors. Web browsers, arguably, are the most exciting place to learn the latest security issues and techniques, but remain as a black art to most security researchers because of their fast-changing characteristics and complex code bases. To bridge this gap, this paper attempts to systematize the security landscape of modern web browsers by studying the popular classes of security bugs, their exploitation techniques, and deployed defenses. More specifically, we first introduce a unified architecture that faithfully represents the security design of four major web browsers. Second, we share insights from a 10-year longitudinal study on browser bugs. Third, we present a timeline and context of mitigation schemes and their effectiveness. Fourth, we share our lessons from a full-chain exploit used in 2020 Pwn2Own competition. and the implication of bug bounty programs to web browser security. We believe that the key takeaways from this systematization can shed light on how to advance the status quo of modern web browsers, and, importantly, how to create secure yet complex software in the future.

cs.CR↗

arXiv2021

Ramanujan and Koshliakov Meet Abel and Plana

The neglected Russian mathematician, N.~S.~Koshliakov, derived beautiful generalizations of the classical Abel--Plana summation formula through a setting arising from a boundary value problem in heat conduction. When we let the parameter $p$ in this setting tend to infinity, his formulas reduce to the classical Abel--Plana summation formula. Rigorous formulations and proofs of these summation formulas are given. In his notebooks, Ramanujan derived different analogues of the Abel--Plana summation formula. One particular example provides a vast new generalization of the classical transformation formula for Eisenstein series, which we generalize in Koshliakov's setting.

math.NT↗

arXiv2021

Recent CMS results on soft QCD physics

Soft quantum chromodynamics (QCD) measurements play an important role in fundamental QCD studies as well as in tuning of corresponding Monte Carlo generator models for a good description of experimental data. Recent results of soft QCD measurements with the CMS experiment, such as minimum bias/underlying event physics, double parton scattering and forward jet production production are presented.

hep-ex↗

arXiv2021

A Class of Identities Associated with Dirichlet Series Satisfying Hecke's Functional Equation

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series of the forms $$\sum_{n=1}^{\infty}\frac{a(n)}{λ_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{μ_n^{s}},$$ satisfying a familiar functional equation involving the gamma function $Γ(s)$. A general identity is established. Appearing on one side is an infinite series involving $a(n)$ and modified Bessel functions $K_ν$, wherein on the other side is an infinite series involving $b(n)$ that is an analogue of the Hurwitz zeta function. Seven special cases, including $a(n)=τ(n)$ and $a(n)=r_k(n)$, are examined, where $τ(n)$ is Ramanujan's arithmetical function and $r_k(n)$ denotes the number of representations of $n$ as a sum of $k$ squares. Most of the six special cases appear to be new.

math.NT↗