The Analytic Strong Multiplicity One Theorem for GL_{m}(A_{K})

Let $\pi=\otimes\pi_{v}$ and $\pi^{\prime}=\otimes\pi_{v}^{\prime}$ be two irreducible, automorphic, cuspidal representations of $GL_{m}(\mathbb{A}_{K}) >.$ Using the logarithmic zero-free region of Rankin-Selberg $L$-function, Moreno established the analytic strong multiplicity one theorem if at least one of them is self-contragredient, i.e. $\pi$ and $\pi^{\prime}$ will be equal if they have finitely many same local components $\pi_{v},\pi_{v}^{\prime},$ for which the norm of places are bounded polynomially by the analytic conductor of these cuspidal representations. Without the assumption of the self-contragredient for $\pi,\pi^{\prime},$ Brumley generalized this theorem by a a different method, which can be seen as an invariant of Rankin-Selberg method. In this paper, influenced by Landau's smooth method of Perron formula, we improved the degree of Brumley's polynomial bound to be $4m+\epsilon.$