arXiv subjects

Ammar Ali Neamah

Publications and source records attributed to Ammar Ali Neamah.

3 recordsLinked to original sources

arXiv2021

Multiplicative functions with sum zero

$CMO$ functions are completely multiplicative functions $f$ for which $\sum_{n=1}^\infty f(n)$ $=0$. These functions were first introduced and studied by Kahane and Saïas [5]. The main purpose of this paper is to generalise such functions to multiplicative functions and we shall call them $MO$ functions. More precisely, we define $MO$ functions to be multiplicative functions for which $\sum_{n=1}^\infty f(n) = 0$ and $\sum_{k=0}^\infty f(p^k)$ $\ne 0$ for all $p \in \mathbb{P}$. We give some properties and find examples of $MO$ functions, as well as pointing out the connection between these functions and the Riemann hypothesis at the end of the paper.

math.NT↗

arXiv2019

The average order of the Möbius function for Beurling primes

In this paper, we study the counting functions $ψ_\mathcal{P}(x)$, $N_\mathcal{P}(x)$ and $M_\mathcal{P}(x)$ of a generalized prime system $\mathcal{N}$. Here $M_\mathcal{P}(x)$ is the partial sum of the Möbius function over $\mathcal{N}$ not exceeding $x$. In particular, we study these when they are asymptotically well-behaved, in the sense that $ψ_{\cal{P}}(x) = x+O({x^{ α+ε}})$, $N_{\cal{P}}(x) = ρx+O({x^{ β+ε}})$ and $ M_\mathcal{P}(x) = O(x^{γ+ε})$, for some $ρ>0$ and $α, β, γ<1$. We show that the two largest of $α,β,γ$ must be equal and at least $\frac{1}{2}$.

math.NT↗

arXiv2016

New Collisions to Improve Pollard's Rho Method of Solving the Discrete Logarithm Problem on Elliptic Curves

It is true that different approaches have been utilised to accelerate the computation of discrete logarithm problem on elliptic curves with Pollard's Rho method. However, trapping in cycles fruitless will be obtained by using the random walks with Pollard's Rho. An efficient alternative approach that is based on new collisions which are reliant on the values ai , bi to solve this problem is proposed. This may requires less iterations than Pollard's Rho original in reaching collision. Thus, the performance of Pollard's Rho method is more efficiently because the improved method not only reduces the number of mathematical operations but these collisions can also applied on previous improvements which reported in the literature.

cs.CR↗