1 5 N ov 2 00 6 The Analytic Strong Multiplicity One Theorem for

Let π = ⊗πv and π ′ = ⊗π ′ v be two irreducible, automorphic, cuspidal representations of GLm (AK). Using the logarithmic zero-free region of Rankin-Selberg L-function, Moreno established the analytic strong multi-plicity one theorem if at least one of them is self-contragredient, i.e. π and π ′ will be equal if they have finitely many same local components πv, π ′ v , 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 π, π ′ , Brumley generalized this theorem by 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 + ε.