1 3 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 Brumley's polynomial bound in the exponent of the analytic conductor to the linear bound of 4m + ε.