arXiv subjects

Roupam Ghosh

Publications and source records attributed to Roupam Ghosh.

6 recordsLinked to original sources

arXiv2015

The Dirichlet series that generates the Möbius function is the inverse of the Riemann zeta function in the right half of the critical strip

In this paper I introduce a criterion for the Riemann hypothesis, and then using that I prove $\sum_{k=1}^\infty μ(k)/k^s$ converges for $\Re(s) > \frac{1}{2}$. I use a step function $ν(x) = 2\{x/2\} - \{x\}$ for the Dirichlet eta function ($\{x\}$ is the fractional part of $x$), which was at the core of my investigations, and hence derive the stated result subsequently.

math.GM↗

arXiv2013

On integrals of fractional parts and the theory of prime differences

In this paper we study the integrals of fractional parts of given functions, and develop some new tools to understand the behaviour of prime differences. We demonstrate how simply some seemingly difficult conjectures related to prime differences can be dealt with. Some, good results discussed here includes, the well know conjecture on prime gaps by Cramér and $\lim \inf_{n\to\infty} d_n < \infty$. Based on some simple assumptions, we have demonstrated how to tackle such problems.

math.GM↗

arXiv2011

A new criterion of the Riemann hypothesis

We introduce a new criterion which if satisfied implies the Riemann hypothesis.

math.GM↗

arXiv2011

An equation satisfied by all non-trivial zeros $ρ$ of the Riemann zeta function $ζ$

We show that if $ρ$ is a non-trivial zero of the Riemann zeta function $ζ$ then $$2^ρ+ \frac{1}{ρ- 1} + 1/2 = ρ\int_{1}^{\infty} {t + 1/2} t^{-ρ-1} dt$$ where, ${x}$ is the fractional part of $x$.

math.GM↗

arXiv2009

New lower bounds for the size of a non-trivial loop in the Collatz 3x+1 and generalized px+q problem

In the Collatz 3x+1 problem, there are 3 possibilities: Starting from any positive number, we either reach the trivial loop (1,4,2), end up in a non-trivial loop, or go until infinity. In this paper, we shall show that if a non-trivial loop with m odd numbers exists, then its minimum odd number is bounded above by a function of m. We shall also use that bound to calculate the least number of odd elements required for a non-trivial loop to exist. Also, the generalized bounds for the px+q problem are given.

math.GM↗

arXiv2008

On the Collatz Problem

Taking a new approach towards analyzing the Collatz Problem, or, 3x+1 conjecture. Introducing some new functions, the Collatz-2 and Collatz-3 sequences, as well as deducing results related to Collatz-2 and Collatz-3 sequences.

math.GM↗