We prove that the uniform \( abc\)-conjecture for number fields implies that there are no Siegel zeros for \( L\)-function of quadratic characters \( (-d/.)\), by studying the singular moduli that give rise to solutions of \( j(\tau)=\gamma_2(\tau)^3=\gamma_3(\tau)^2+1728\).

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We show that for every fundamental discriminant \( D>3705\) there is a prime \(p<\sqrt{D}/2\) for which \( (D/p)=-1\).

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We show that there is a constant \( c\in (\frac 12,1)\) such that \( \sim cp\) zeros of the \( p\)th Fekete polynomial lie on the unit circle

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We bound from below the lcm of \( k\) integers from a short interval. This is then used to bound the length of any arc of the hyperbola \( xy=N\) which contains \( k\) lattice points.

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