arXiv subjects
Publications and source records attributed to Masum Billal.
In a paper in the American Mathematical Monthly, the corresponding author asks for an asymptotic of a gcd-sum function \begin{align}\sum_{ab\leq N}τ(\gcd(a,b))\label{eqn:taugcdsum}\end{align} We extensively study generalizations of the sum in the aforementioned paper and establish asymptotic results using elementary methods only.
The purpose of this memoir is to discuss two very interesting properties of integer sequences. One is the law of apparition and the other is the law of repetition. Both have been extensively studied by mathematicians such as Ward, Lucas, Lehmer, Hall, etc. However, due to the lack of a proper survey in this area, many results have been rediscovered many decades later. This along with the necessity of the usefulness of such theory calls for a survey on this topic.
The purpose of this paper is to investigate integer sequences with exponent lifting property, a property common in Fibonacci or Lucas sequences.
The number of tuples with positive integers pairwise relatively prime to each other with product at most $n$ is considered. A generalization of $μ^{2}$ where $μ$ is the Möbius function is used to formulate this divisor sum and establish some identities. One such identity is a weighted sum of reciprocal of square-free numbers not exceeding $n$. Some auxiliary number theoretic functions are introduced to formulate this sum.
We dedicate this paper to investigate the most generalized form of Fibonacci Sequence, one of the most studied sections of the mathematical literature. One can notice that, we have discussed even a more general form of the conventional one. Although it seems the topic in the first section has already been covered before, but we present a different proof here. Later I found out that, the auxiliary theorem used in the first section was proven and even generalized further by F. T. Howard. Thanks to Curtis Cooper for pointing out the fact that this has already been studied and providing me with references. the At first, we prove that, only the common general Fibonacci Sequence can be a divisible sequence under some restrictions. In the latter part, we find some properties of the sequence, prove that there are infinite alternating bisquable Fibonacci sequence(defined later) and provide a lower bound on the number of divisors of Fibonacci numbers.