 
Summary: NotesontheDeuring
Heilbronn
Phenomenon
Jeffrey Stopple
864 NOTICES OF THE AMS VOLUME 53, NUMBER 8
Introduction
Analytic number theory studies Lfunctions, gen
eralizations of the Riemann zeta function (s). It
can be difficult to see why this is number theory.
In fact, the Class Number Formula (6) of Dirichlet
gives the number h(d) of classes of binary qua
dratic forms of discriminant d as the value of such
an Lfunction at s = 1. The location of the zeros is
important: since the functions are continuous, the
value is influenced by any zero of the function
near s = 1. Such a zero would of course contradict
the Generalized Riemann Hypothesis (GRH).
The DeuringHeilbronn phenomenon says that
such a counterexample to the GRH for one L
function would influence the horizontal and ver
