Document Type : Research Paper

Authors

• Nasir Uddin Gazi ¹
• Rana Mondal ²

¹ Department of Mathematics and Statistics, Aliah University, IIA/27, AA II, Newtown, Kolkata-700160, India

² Department of Mathematics, Cambridge Institute of Technology, Ranchi, 835103, Jharkhand, India

Abstract

 $L$-functions are complex functions associated with number-theoretic objects such as number fields, elliptic curves, modular forms, and automorphic representations. The general form of an $L$-function can be represented as a Dirichlet series, an Euler product, or in terms of its analytic continuation and functional equation. One of the most famous $L$-functions is the Riemann zeta function, defined as: $\zeta (s)=1^s+2^{-s}+3^{-s}+\cdots =\sum _{n=1}^{\mathrm{\infty }}{n}^{-s}$, where s is a complex number. $L$- function plays a fundamental role in studying prime numbers and connects to important conjectures like the Riemann Hypothesis. In this paper, we study the uniqueness of transcendental meromorphic functions and $L$-function whose certain difference-differential polynomials share a small function and rational function with weight, where $L$-function is a function that is Dirichlet series with the Riemann zeta function as the prototype. The Selberg class $S$ of $L$-functions is the set of all Dirichlet series $L(s)=\sum _{n=1}^{\mathrm{\infty }}a(n)n^{-s}$ of a complex variable $s=\sigma +it$ with $a(1)=1$.

Keywords

• Uniqueness
• Meromorphic functions
• Entire functions
• L-function
• Sharing values