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On Landau-Siegel zeros and heights of singular moduli
Christian Táfula
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Let χ_D be the Dirichlet character associated to ℚ(√(D)) where D < 0 is a fundamental discriminant. Improving Granville-Stark [DOI:10.1007/s002229900036], we show that (L′)/L(1,χ_D) = 1/6 height(j(τ_D)) - 1/2log|D| + C + o_(D→-∞)(1), where τ_D = 1/2(-δ+√(D)) for D ≡δ (mod 4) and j(·) is the j-invariant function with C = -1.057770…. Assuming the ``uniform'' abc-conjecture for number fields, we deduce that L(β,χ_D)0 with β≥1 - (√(5)φ+ o(1))/(log|D|) where φ= (1+√(5))/2, which we improve for smooth D.
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