9 research outputs found
The first factor of the class number of the -th cyclotomic field
Kummer's conjecture states that the relative class number of the -th
cyclotomic field follows a strict asymptotic law. Granville has shown it
unlikely to be true -- it cannot be true if we assume the truth of two other
widely believed conjectures. We establish a new bound for the error term in
Kummer's conjecture, and more precisely we prove that
, where is a possible
Siegel zero of an , odd.Comment: 8 page
Dimensions of the irreducible representations of the symmetric and alternating group
We establish the existence of an irreducible representation of whose
dimension does not occur as the dimension of an irreducible representation of
, and vice versa. This proves a conjecture by Tong-Viet. The main
ingredient in the proof is a result on large prime factors in short intervals.Comment: 24 page
Asymptotic expansions, -values and a new Quantum Modular Form
In 2010 Zagier introduced the notion of a quantum modular form. One of his
first examples was the "strange" function of Kontsevich. Here we produce
a new example of a quantum modular form by making use of some of Ramanujan's
mock theta functions. Using these functions and their transformation behaviour,
we also compute asymptotic expansions similar to expansions of .Comment: 7 page
Irregular behaviour of class numbers and Euler-Kronecker constants of cyclotomic fields: the log log log devil at play
Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on
invariants related to the cyclotomic field with a
prime. Kummer's conjecture concerns the asymptotic behaviour of the first
factor of the class number of and Ihara's the positivity
of the Euler-Kronecker constant of (the ratio of the
constant and the residue of the Laurent series of the Dedekind zeta function
at ). If certain standard conjectures in
analytic number theory hold true, then one can show that both conjectures are
true for a set of primes of natural density 1, but false in general.
Responsible for this are irregularities in the distribution of the primes. With
this survey we hope to convince the reader that the apparently dissimilar
mathematical objects studied by Kummer and Ihara actually display a very
similar behaviour.Comment: 20 pages, 1 figure, survey, to appear in `Irregularities in the
Distribution of Prime Numbers - Research Inspired by Maier's Matrix Method',
Eds. J. Pintz and M. Th. Rassia
Dimensions of the irreducible representations of the symmetric and alternating group
We establish the existence of an irreducible representation of A_n whose dimension does not occur as the dimension of an irreducible representation of S_n, and vice versa. This proves a conjecture by Tong-Viet. The main ingredient in the proof is a result on large primefactors in short interval