Math @ Duke

Publications [#9773] of Joshua B. Holden
Papers Submitted
 Joshua Brandon Holden, Firsthit analysis of algorithms for computing quadratic irregularity,
Mathematics of Computation
, submitted 2000 [math.NT/0011054]
(last updated on 2000/12/22)
Abstract: The author has previously extended the theory of regular and
irregular
primes to the setting of arbitrary totally real number
fields. It has been
conjectured that the Bernoulli numbers, or alternatively the
values of the
Riemann zeta function at odd negative integers, are evenly
distributed
modulo p for every p. This is the basis of a wellknown
heuristic
given by Siegel for estimating the frequency of irregular
primes. So
far, analyses have shown that if Q(\sqrt{D}) is a real
quadratic field,
then the values of the zeta function
\zeta_{D}(12m)=\zeta_{Q(\sqrt{D})}(12m) at negative odd
integers are
also distributed as expected modulo p for any p. We use
this
heuristic to predict the computational time required to find
quadratic
analogues of irregular primes with a given order of
magnitude. We also
discuss alternative ways of collecting large amounts of data
to test the
heuristic.


dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821
 
Mathematics Department
Duke University, Box 90320
Durham, NC 277080320

