Andrew Granville's Home Page

2000 Publications

ABC implies no "Siegel Zeroes'' for L-functions of characters with negative discriminant (with Harold Stark)
Inventiones Mathematicae, 139 (2000), 509-523.

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 \( j(\tau)=\gamma_2(\tau)^3=\gamma_3(\tau)^2+1728\).


An upper bound on the least inert prime in a real quadratic field (with Richard Mollin and Hugh C. Williams)
Canadian Journal of Mathematics, 52 (2000), 369--380.

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


Zeros of Fekete polynomials (with Brian Conrey, K. Soundararajan and Bjorn Poonen)
Annales de l'Institut Fourier (Grenoble), 50 (2000), 865--889.

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


The least common multiple and lattice points on hyperbolas (with Jorge Jiménez-Urroz)
Quarterly Journal of Mathematics (Oxford), 51 (2000), 343--352.

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.