# COUNTING $S_5$ -FIELDS WITH A POWER SAVING ERROR TERM

@article{Shankar2014COUNTING,
title={COUNTING \$S\_5\$ -FIELDS WITH A POWER SAVING ERROR TERM},
author={Arul Shankar and Jacob Tsimerman},
journal={Forum of Mathematics, Sigma},
year={2014},
volume={2}
}
• Published 8 October 2013
• Mathematics
• Forum of Mathematics, Sigma
Abstract We 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.
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## References

SHOWING 1-10 OF 15 REFERENCES
Error estimates for the Davenport-Heilbronn theorems
• Mathematics
• 2010
We obtain the first known power-saving remainder terms for the theorems of Davenport and Heilbronn on the density of discriminants of cubic fields and the mean number of 3-torsion elements in the
The density of discriminants of quartic rings and fields
We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a
Analytic Number Theory
• Mathematics
• 2004
Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large
The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
• Mathematics
• 2012
We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It
Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
• Mathematics
• 2010
We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and
Most hyperelliptic curves over Q have no rational points
By a hyperelliptic curve over Q, we mean a smooth, geometrically irreducible, complete curve C over Q equipped with a fixed map of degree 2 to P^1 defined over Q. Thus any hyperelliptic curve C over
Higher composition laws IV: The parametrization of quintic rings
rst three parts of this series, we considered quadratic, cubic and quartic rings (i.e., rings free of ranks 2, 3, and 4 over Z) respectively, and found that various algebraic structures involving
Error terms for the Davenport–Heilbronn theorems
• Duke Math. J. 153
• 2010