Abstract
AbstractWe show how the Selberg $\Lambda ^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using the geometry of numbers. Specifically, we give such an error term for the counting function of $S_5$-quintic fields.
Highlights
Over the past decade there has emerged a large body of work concerned with counting arithmetic objects by parameterizing them as GZ orbits on VZ, where G is some reductive algebraic group, and V is a representation of G
We show how the Selberg Λ2-sieve can be used very generally to obtain such power savings
We demonstrate our claim by obtaining the first known power saving for quintic fields with Galois group S5 and bounded discriminant: c The Author(s) 2014
Summary
Over the past decade there has emerged a large body of work concerned with counting arithmetic objects by parameterizing them as GZ orbits on VZ, where G is some reductive algebraic group, and V is a representation of G (see [3, 5,6,7,8,9, 11]). We show how the Selberg Λ2-sieve can be used very generally to obtain such power savings. We demonstrate our claim by obtaining the first known power saving for quintic fields with Galois group S5 and bounded discriminant: c The Author(s) 2014.
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