Math @ Duke

Publications [#9774] of Joshua B. Holden
Papers Accepted
 Joshua Brandon Holden, Distribution of Values of Real Quadratic Zeta Functions,
Proceedings of the DIMACS Workshop on Unusual Applications of Number Theory
, accepted 2000 [math.NT/0010285]
(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. However,
it has
proven to be very computationally intensive to calculate
these numbers for
large values of m. In this paper, we present the
alternative of
computing \zeta_{D}(12m) for a fixed value of D and a large
number of
different m.


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