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

Effective Lower Bounds for L(1,{\chi}) via Eisenstein Series

We give effective lower bounds for $L(1,\chi)$ via Eisenstein series on $\Gamma_0(q) \backslash \mathbb{H}$. The proof uses the Maass-Selberg relation for truncated Eisenstein series and sieve theory in the form of the Brun-Titchmarsh inequality. The method follows closely the work of Sarnak in using Eisenstein series to find effective lower bounds for $\zeta(1+it)$.Comment: 17 pages. To appear in Pacific Journal of Mathematic

The Mertens and Pólya conjectures in function fields

The Mertens conjecture on the order of growth of the summatory function of the M{u00F6}bius function has long been known to be false. We formulate an analogue of this conjecture in the setting of global function fields, and investigate the plausibility of this conjecture. First we give certain conditions, in terms of the zeroes of the associated zeta functions, for this conjecture to be true. We then show that in a certain family of function fields of low genus, the average proportion of curves satisfying the Mertens conjecture is zero, and we hypothesise that this is true for any genus. Finally, we also formulate a function field version of P{u00F3}lya's conjecture, and prove similar results

Biases in prime factorizations and Liouville functions for arithmetic progressions

We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arithmetic progression in the prime factorizations of integers. For example, we observe that the primes of the form $4k+1$ tend to appear an even number of times in the prime factorization of a given integer, more so than for primes of the form $4k+3$. We are led to consider variants of P\'olya's conjecture, supported by extensive numerical evidence, and its relation to other conjectures.Comment: 25 pages, 6 figure