Assuming the generalized Ramanujan conjecture, ∫_T^{2T} |L(1/2+it, π)|^2 dt ≪_π T^{d/2} / log^{η_d} T holds for small η_d > 0 when d ≥ 3.
A Brun-Titchmarsh theorem for multiplicative func tions
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On the second integral moment of $L$-functions
Assuming the generalized Ramanujan conjecture, ∫_T^{2T} |L(1/2+it, π)|^2 dt ≪_π T^{d/2} / log^{η_d} T holds for small η_d > 0 when d ≥ 3.