807 research outputs found
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 via Eisenstein series on
. 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 .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 tend to
appear an even number of times in the prime factorization of a given integer,
more so than for primes of the form . 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
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