Ankeny–Artin–Chowla congruence について

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Ankeny–Artin–Chowla congruence ：ウィキペディア英語版

Ankeny–Artin–Chowla congruence
In number theory, the Ankeny–Artin–Chowla congruence is a result published in 1953 by N. C. Ankeny, Emil Artin and S. Chowla. It concerns the class number ''h'' of a real quadratic field of discriminant ''d'' > 0. If the fundamental unit of the field is
:

\varepsilon = \frac

with integers ''t'' and ''u'', it expresses in another form
:

\frac \pmod\;

for any prime number ''p'' > 2 that divides ''d''. In case ''p'' > 3 it states that
:

-2 \equiv \sum_ \lfloor  \rfloor \pmod

where

m = \frac\;

and

\chi\;

is the Dirichlet character for the quadratic field. For ''p'' = 3 there is a factor (1 + ''m'') multiplying the LHS. Here
:

\lfloor x\rfloor

represents the floor function of ''x''.
A related result is that if ''d=p'' is congruent to one mod four, then
:

h \equiv B_ \pmod

where ''B''_''n'' is the ''n''th Bernoulli number.
There are some generalisations of these basic results, in the papers of the authors.
==References==

*

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