Weighted sharing and uniqueness of L-Function with certain class of meromorphic function

Abstract: The purpose of the paper is to study the uniqueness problem of a $L$ function in the Selberg class sharing one or two sets with an arbitrary meromorphic function having finite poles. We manipulate the notion of weighted sharing of sets to improve one result of Yuan-Li-Yi [Value distribution of $L$-functions and uniqueness questions of F. Gross, Lithuanian Math. J., $\mathbf{58(2)}$(2018), 249-262]. More importantly, we have pointed out a number of gaps in all the results of Sahoo-Halder [Results on $L$ functions and certain uniqueness question of Gross, Lithuanian Math. J., $\mathbf{60(1)}$(2020), 80-91] which actually makes the validity of the same paper under question. As an attempt to rectify the results of Sahoo-Halder we have presented the accurate forms and proof of the results in a compact and convenient manner.