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arXiv:1606.06103 (Published 2016-06-20)
On $\ell$-torsion in class groups of number fields
Comments: 25 pagesCategories: math.NTFor each integer $\ell \geq 1$, we prove an unconditional upper bound on the size of the $\ell$-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of $\mathbb{Q}$ of degree $d$, for any fixed $d \in \{2,3,4,5\}$ (with the additional restriction in the case $d=4$ that the field be non-$D_4$). For sufficiently large $\ell$ (specified explicitly), these results are as strong as a previously known bound that is conditional on GRH. As part of our argument, we develop a probabilistic "Chebyshev sieve," and give uniform, power-saving error terms for the asymptotics of quartic (non-$D_4$) and quintic fields with chosen splitting types at a finite set of primes.
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arXiv:1410.2927 (Published 2014-10-10)
The sequence of fractional parts of roots
Comments: 16 pagesCategories: math.NTWe study the function M(t,n) = Floor[ 1 / {t^(1/n)} ], where t is a positive real number, Floor[.] and {.} are the floor and fractional part functions, respectively. In a recent article in the Monthly, Nathanson proved that if log(t) is rational, then for all but finitely many positive integers n one has M(t,n) = Floor[ n / log(t) - 1/2 ]. We extend this by showing that, without condition on t, all but a zero-density set of integers n satisfy M(t,n) = Floor[ n / log(t) - 1/2 ]. Using a metric result of Schmidt, we show that almost all t have asymptotically log(t) log(x)/12 exceptional n<x. Using continued fractions, we produce uncountably many t that have only finitely many exceptional n, and also give uncountably many explicit t that have infinitely many exceptional n.