74 research outputs found
Bertrand's postulate and subgroup growth
In this article we investigate the L^1-norm of certain functions on groups
called divisibility functions. Using these functions, their connection to
residual finiteness, and integration theory on profinite groups, we define the
residual average of a finitely generated group. One of the main results in this
article is the finiteness of residual averages on finitely generated linear
groups. Whether or not the residual average is finite depends on growth rates
of indices of finite index subgroups. Our results on index growth rates are
analogous to results on gaps between primes, and provide a variant of the
subgroup growth function, which may be of independent interest.Comment: 33 page
Bertrand's Postulate for Carmichael Numbers
Alford, Granville, and Pomerance proved that there are infinitely many
Carmichael numbers. In the same paper, they ask if a statement analogous to
Bertrand's postulate could be proven for Carmichael numbers. In this paper, we
answer this question, proving the stronger statement that for all
and sufficiently large in terms of , there exist at least
Carmichael numbers between and
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