arXiv subjects

Nilotpal Kanti Sinha

Publications and source records attributed to Nilotpal Kanti Sinha.

9 recordsLinked to original sources

arXiv2026

Chebyshev Bias for Largest Prime Factors

Let $P^+(n)$ be the largest prime factor of $n$, and let $χ=χ_{-4}$ be $1$ on primes $1\pmod4$ and $-1$ on primes $3\pmod4$. For fixed $k\ge2$ we study \[ D_k(x)=\sum_{\substack{n\le x\\ Ω(n)=k}}χ(P^+(n)). \] Thus $D_k(x)$ compares the two residue classes according to the largest prime factor of integers having exactly $k$ prime factors, counted with multiplicity. Assuming RH for $ζ(s)$ and $L(s,χ)=β(s)$, we prove a pointwise explicit formula. The main term is a fixed negative contribution plus an absolutely convergent oscillating sum over the zeros $ρ=\frac12+iγ$ of $L(s,χ)$. The ordering of the prime factors produces $k$ Perron denominators, and the principal coefficient of a zero is $O_k((1+|γ|)^{-k})$. We then show that the total size of all zero terms is strictly smaller than the fixed contribution. Hence $D_k(x)<0$ for all sufficiently large $x$. The analogous problem without fixing $k$ is still open.

math.GM↗

arXiv2026

Asymptotic Formulae For Reciprocal Partitions

For integers $1\le m\le n$, let $s(n,m)$ denote the number of $m$-tuples $(k_1,\ldots,k_m)$ of nonnegative integers satisfying \[ n=\sum_{j=1}^{m}\frac{k_j}{j}. \] We obtain a complete asymptotic expansion for $\log s(n,m)$, uniformly for all $n\ge m$ as $m\to\infty$. The coefficients are given explicitly in terms of limiting prime-block functions arising from the residue structure of the problem. We also determine the full hierarchy of multiplicative corrections in the sparse regime, identifying explicit constants at every fixed order and, in particular, the first correction constants $1/4$ and $(1-\log 2)/4$. The results give uniform two-parameter asymptotics as both the target and the number of allowed reciprocal parts grow.

math.GM↗

arXiv2025

Accelerating Photonic Integrated Circuit Design: Traditional, ML and Quantum Methods

Photonic Integrated Circuits (PICs) provide superior speed, bandwidth, and energy efficiency, making them ideal for communication, sensing, and quantum computing applications. Despite their potential, PIC design workflows and integration lag behind those in electronics, calling for groundbreaking advancements. This review outlines the state of PIC design, comparing traditional simulation methods with machine learning approaches that enhance scalability and efficiency. It also explores the promise of quantum algorithms and quantum-inspired methods to address design challenges.

quant-ph↗

arXiv2024

Ultralight Signal Classification Model for Automatic Modulation Recognition

The growing complexity of radar signals demands responsive and accurate detection systems that can operate efficiently on resource-constrained edge devices. Existing models, while effective, often rely on substantial computational resources and large datasets, making them impractical for edge deployment. In this work, we propose an ultralight hybrid neural network optimized for edge applications, delivering robust performance across unfavorable signal-to-noise ratios (mean accuracy of 96.3% at 0 dB) using less than 100 samples per class, and significantly reducing computational overhead.

cs.LG↗

arXiv2021

A general model of regression using iterative series

We present a new and general method of weighted least square univariate regression where the dependent variable is expanded as a series of suitably chosen functions of the independent variables. Each term of the series is obtained by an iterative process which reduces the sum of the square of the residuals. Thus by evaluating the regression series to a sufficiently large number of terms we can, in principle, reduce the sum of the square of residuals and improve the accuracy of the fit.

math.NA↗

arXiv2015

On the asymptotic expansion of the sum of the first n primes

An asymptotic formula for the sum of the first n primes is derived. This result improves the previous results of P. Dusart. Using this asymptotic expansion, we prove the conjectures of R. Mandl and G. Robin on the upper and the lower bound of the sum of the first n primes respectively.

math.NT↗

arXiv2012

On the half line: K. Ramachandra

A short biographical note on the life and works of K. Ramachandra, one of the leading mathematicians in the field of analytic number theory in the second half of the twentieth century.

math.HO↗

arXiv2011

On a unified theory of numbers

We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be independent results can be unified under a single equation. We apply our method to derive several new results in the above areas of number theory.

math.NT↗

arXiv2010

On a new property of primes that leads to a generalization of Cramer's conjecture

I present a new property of prime numbers that leads to a generalization of Cramer's conjecture. The study of the gap between consecutive primes is treated as a special case of the gap between consecutive terms of sequences having a certain property called pseudo equidistribution. Both rigorous as well as heuristic arguments are given in support of the generalized Cramer's conjectures.

math.NT↗