Aggregate Zero Density Bounds for a Family of Automorphic L-functions
In this paper,aggregate zero density bounds for a family of automorphic L-functions are deduced from bounds for a sum of integral power moments of such L-functions.More precisely,let I be a set of certain automorphic representations π,and let c(π)be a non-negative coefficient for each π ∈ I such that Σπ∈I c(π)converges.Assume that∑π∈Ic(π)∫T+TαT|L(1/2+it,π)|2ℓdt＜＜εTθ+ε∑π∈Ic(π)for certain ℓ≥1,0＜α≤1 and θ ≥ α.Upper bounds for the following aggregate zero density∑π∈Ic(π)Nπ(σ,T,T+Tα)will be proved,where Nπ(σ,T1,T2)is the number of zeros ρ=β+iγ of L(s,π)inσ＜β＜1 and T1≤γ≤ T2.
zero densityaggregate zero densityRiemann zeta-functionautomorphic L-functionintegral power momentaggregate integral power moment
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