Equipolar meromorphic functions sharing a set

"Two meromorphic functions f and g having the same set of poles are known as equipolar. In this paper we study some uniqueness results of equi-polar meromorphic functions sharing a finite set and improve some recent results of Bhoosnurmath-Dyavanal [4] and Banerjee-Mallick [3] by removing some unnec- essary conditions on rami cation indices as well as relaxing the condition on the nature of sharing of the value infinity by f and g from counting multiplicity to ignoring multiplicity."

On the proximity of multiplicative functions to the number of distinct prime factors function

Given an additive function f and a multiplicative function g, let E(f, g;x) = #{n ≤ x: f(n) = g(n)}. We study the size of E(ω,g;x) and E(Ω,g;x), where ω(n) stands for the number of distinct prime factors of n and Ω(n) stands for the number of prime factors of n counting multiplicity. In particular, we show that E(ω,g;x) and E(Ω,g;x) are $\begin{array}{} \displaystyle O\left(\frac{x}{\sqrt{\log\log x}}\right) \end{array}$ for any integer valued multiplicative function g. This improves an earlier result of De Koninck, Doyon and Letendre.

Algebraic dependences of meromorphic mappings sharing moving hyperplanes without counting multiplicities

This paper deals with the multiple values and algebraic dependences problem of meromorphic mappings sharing moving hyperplanes in projective space. We give some algebraic dependences theorems for meromorphic mappings sharing moving hyperplanes without counting multiplicity, where all zeros with multiplicities more than a certain number are omitted. Basing on these results, some unicity theorems regardless of multiplicity for meromorphic mappings in several complex variables are given. These results are extensions and strong improvements of some recent results.

Uniqueness of Differential Polynomials of Meromorphic Functions Sharing a Small Function Without Counting Multiplicity

The paper concerns interesting problems related to the field of Complex Analysis, in particular Nevanlinna theory of meromorphic functions. The author have studied certain uniqueness problem on differential polynomials of meromorphic functions sharing a small function without counting multiplicity. The results of this paper are extension of some problems studied by K. Boussaf et. al. in [2] and generalization of some results of S.S. Bhoosnurmath et. al. in [4].

Uniqueness of difference-differential polynomials of entire functions sharing one value

In this paper, we study the uniqueness of difference-differential polynomials of entire functions $f$ and $g$ sharing one value with counting multiplicity. In this paper we extend and generalize the results of X. Y. Zhang, J. F. Chen and W. C. Lin [17] L. Kai, L. Xin-ling and C. Ting-bin [7] and many others [2, 16].

On the residue class distribution of the number of prime divisors of an integer

Let Ω(n) denote the number of prime divisors of n counting multiplicity. One can show that for any positive integer m and all j = 0,1,…,m – 1, we havewith α = 1. Building on work of Kubota and Yoshida, we show that for m > 2 and any j = 0,1,…,m – 1, the error term is not o(xα) for any α < 1.

On the residue class distribution of the number of prime divisors of an integer

Let Ω(n) denote the number of prime divisors ofncounting multiplicity. One can show that for any positive integermand allj= 0,1,…,m– 1, we havewithα= 1. Building on work of Kubota and Yoshida, we show that form&gt; 2 and anyj= 0,1,…,m– 1, the error term is noto(xα) for anyα&lt; 1.