**
Zhenchao
Ge,
Department
of
Pure
Mathematics,
University
of
Waterloo**

**
"Irregularities
of
Dirichlet
L-functions
and
a
parity
bias
in
gaps
of
zeros"**

The
integral
of
Hardy's
Z-function
from
$0$
to
$T$
measures
the
occurrence
of
its
sign
changes.
Hardy
proved
that
this
integral
is
$o(T)$
from
which
he
deduced
that
the
Riemann
zeta-function
has
infinitely
many
zeros
on
the
critical
line.
A.
Ivić conjectured
this
integral
is
$O(T^{1/4})$
and
$\Omega_{\pm}(T^{1/4})$
as
$T\to\infty$.
These
estimates
were
proved,
independently,
by
M.
A.
Korolev
and
M.
Jutila.

In
this
talk,
we
will
show
that
the
analogous
conjecture
is
false
for
the
Z-functions
of
certain
"special"
Dirichlet
L-functions.
In
particular,
we
show
that
the
integral
of
the
Z-function
of
a
Dirichlet
L-functions
from
$0$
to
$T$
is
asymptotic
to
$c_\chi
T^{3/4}$
and
we
classify
precisely
when
the
constant
$c_\chi$
is
nonzero.
Somewhat
surprisingly,
numerical
evidence
seems
to
suggest
that
the
unexpectedly
large
mean
value
is
caused
by
a
currently
unexplained
parity
bias
in
the
gaps
between
the
zeros
of
these
"special"
Dirichlet
L-functions.

This
is
joint
work
with
Jonathan
Bober
and
Micah
Milinovich.

MC 5501