13 research outputs found
An effective Chebotarev density theorem for families of fields, with an application to class groups
This talk will present an effective Chebotarev theorem that holds for all but a possible zero-density subfamily of certain families of number fields of fixed degree. For certain families, this work is unconditional, and in other cases it is conditional on the strong Artin conjecture and certain conjectures on counting number fields. As an application, we obtain nontrivial average upper bounds on β-torsion in the class groups of the families of fields
Consecutive primes in tuples
In a recent advance towards the Prime -tuple Conjecture, Maynard and Tao
have shown that if is sufficiently large in terms of , then for an
admissible -tuple of linear forms in
, the set contains at
least primes for infinitely many . In this note, we
deduce that contains at least
consecutive primes for infinitely many . We answer an old
question of Erd\H os and Tur\'an by producing strings of consecutive
primes whose successive gaps form an increasing
(resp. decreasing) sequence. We also show that such strings exist with
for . For any coprime integers
and we find arbitrarily long strings of consecutive primes with bounded
gaps in the congruence class .Comment: Revised versio
An effective Chebotarev density theorem for families of number fields, with an application to -torsion in class groups
We prove a new effective Chebotarev density theorem for Galois extensions
that allows one to count small primes (even as small as an
arbitrarily small power of the discriminant of ); this theorem holds for the
Galois closures of "almost all" number fields that lie in an appropriate family
of field extensions. Previously, applying Chebotarev in such small ranges
required assuming the Generalized Riemann Hypothesis. The error term in this
new Chebotarev density theorem also avoids the effect of an exceptional zero of
the Dedekind zeta function of , without assuming GRH. We give many different
"appropriate families," including families of arbitrarily large degree. To do
this, we first prove a new effective Chebotarev density theorem that requires a
zero-free region of the Dedekind zeta function. Then we prove that almost all
number fields in our families yield such a zero-free region. The innovation
that allows us to achieve this is a delicate new method for controlling zeroes
of certain families of non-cuspidal -functions. This builds on, and greatly
generalizes the applicability of, work of Kowalski and Michel on the average
density of zeroes of a family of cuspidal -functions. A surprising feature
of this new method, which we expect will have independent interest, is that we
control the number of zeroes in the family of -functions by bounding the
number of certain associated fields with fixed discriminant. As an application
of the new Chebotarev density theorem, we prove the first nontrivial upper
bounds for -torsion in class groups, for all integers ,
applicable to infinite families of fields of arbitrarily large degree.Comment: 52 pages. This shorter version aligns with the published paper. Note
that portions of Section 8 of the longer v1 have been developed as a separate
paper with identifier arXiv:1902.0200
Gaps between zeros of Dedekind zeta-functions of quadratic number fields. II
Let be a quadratic number field and be the associated
Dedekind zeta-function. We show that there are infinitely many normalized gaps
between consecutive zeros of on the critical line which are
greater than times the average spacing.Comment: 12 pages; to appear in the Quarterly Journal of Mathematic
Benford Behavior of Zeckendorf Decompositions
A beautiful theorem of Zeckendorf states that every integer can be written
uniquely as the sum of non-consecutive Fibonacci numbers . A set is said to satisfy Benford's law if
the density of the elements in with leading digit is
; in other words, smaller leading digits are more
likely to occur. We prove that, as , for a randomly selected
integer in the distribution of the leading digits of the
Fibonacci summands in its Zeckendorf decomposition converge to Benford's law
almost surely. Our results hold more generally, and instead of looking at the
distribution of leading digits one obtains similar theorems concerning how
often values in sets with density are attained.Comment: Version 1.0, 12 pages, 1 figur
Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals
Zeckendorf's theorem states that every positive integer can be written
uniquely as a sum of non-consecutive Fibonacci numbers , with initial
terms . We consider the distribution of the number of
summands involved in such decompositions. Previous work proved that as the distribution of the number of summands in the Zeckendorf
decompositions of , appropriately normalized, converges
to the standard normal. The proofs crucially used the fact that all integers in
share the same potential summands.
We generalize these results to subintervals of as ; the analysis is significantly more involved here as different integers
have different sets of potential summands. Explicitly, fix an integer sequence
. As , for almost all the distribution of the number of summands in the Zeckendorf
decompositions of integers in the subintervals ,
appropriately normalized, converges to the standard normal. The proof follows
by showing that, with probability tending to , has at least one
appropriately located large gap between indices in its decomposition. We then
use a correspondence between this interval and to obtain
the result, since the summands are known to have Gaussian behavior in the
latter interval. % We also prove the same result for more general linear
recurrences.Comment: Version 1.0, 8 page