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

Article

#### 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 $$(D/p)=-1$$.

Article

#### 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

Article

#### 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.

Article