Linear relations of zeroes of the zeta-function

This article considers linear relations between the non-trivial zeroes of the Riemann zeta-function. The main application is an alternative disproof to Mertens' conjecture. We show that $\limsup M(x)x^{-1/2} \geq 1.6383$ and that $\liminf M(x)x^{-1/2}\leq -1.6383$.Comment: 12 pages, 2 figures, 2 tables. Version 2: some typos corrected. To appear in Math. Com

On the Mertens Conjecture for Function Fields

We study an analogue of the Mertens conjecture in the setting of global function fields. Building on the work of Cha, we show that most hyperelliptic curves do not satisfy the Mertens conjecture, but that if we modify the Mertens conjecture to have a larger constant, then this modified conjecture is satisfied by a positive proportion of hyperelliptic curves.Comment: 17 pages. Several minor revisions and corrections based on referee comments. To appear in International Journal of Number Theor

On the Mertens Conjecture for Elliptic Curves over Finite Fields

We introduce an analogue of the Mertens conjecture for elliptic curves over finite fields. Using a result of Waterhouse, we classify the isogeny classes of elliptic curves for which this conjecture holds in terms the size of the finite field and the trace of the Frobenius endomorphism acting on the curve.Comment: 12 pages. Minor revisions and additional references added. To appear in Bulletin of the Australian Mathematical Societ

The notion of dimension in geometry and algebra

This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are invoked and compared.Comment: 29 pages, a revie