Sacha Mangerel, University of Toronto
"Some Applications of Pretentiousness in the Theory of Dirichlet Characters"
"Pretentious" methods
in
analytic
number
theory,
as
introduced
by
Granville
and
Soundararajan,
are
powerful
tools
in
the
analysis of
mean
values
and
correlations
of
multiplicative
functions.
In this
talk,
we
will
give
two
applications
of
these
tools.
i)
For
a
non-principal
Dirichlet
character
$\chi$
modulo
$q$, define
$M(\chi)
=
\max_{t}
|\sum_{n
\leq
t}
\chi(n)|$.
The classical
Polya--Vinogradov
inequality
asserts
that $M(\chi)
\ll
\sqrt{q}
\log
q$
unconditionally,
and
on
the Generalized
Riemann
Hypothesis
(GRH),
Montgomery
and
Vaughan showed
that
$M(\chi)
\ll
\sqrt{q}
\log
\log
q$.
We
discuss
a recent
improvement,
in
joint
work
with
Y. Lamzouri,
to
both
of these
results
in
the
case
that
$\chi$
has
odd
order.
This improvement
on
GRH
is
best
possible
up
to
a
factor
of $\log
\log
\log
\log
q$.
One
of
the
key
ingredients
in
the
proof
of
the
upper
bounds
is
a new
Halasz-type
inequality
for
logarithmic
mean
values
of completely
multiplicative
functions.
ii)
Time
permitting,
we
will
also
discuss
the
following
rigidity theorem
related
to
binary
correlations
of
Dirichlet
characters.
If
$f
:
\mathbb{N}
\rightarrow
\mathbb{C}$
is
a
1-bounded multiplicative
function
for
which
there
is
a
primitive
Dirichlet character
$\chi$
of
conductor
$q$
such
that
$$
\sum_{n
\leq
x}
f(n)
\overline{f(n
+
h)}
=
(1
+
o(1))
\sum_{n
\leq
x}
\chi(n)
\overline{\chi(n
+
h)}
$$
for
all
shifts
$|h|
\leq
H$,
then
if
$H
\rightarrow
\infty$
with
$x$
then
$f$
``pretends''
to
be
$\chi$
in
a
precise
sense.
This
is
joint
work
with
O. Klurman.
MC 5403