Asif Zaman, University of Toronto
"A variant of Brun-Titchmarsh for the Chebotarev density theorem"
The
classical
Brun-Titchmarsh
theorem
provides
an
upper
bound
for
the
number
of
primes
in
an
arithmetic
progression
in
a
far
wider
range
than
that
afforded
by
the
Prime
Number
Theorem
for
arithmetic
progressions.
The
Chebotarev
density
theorem,
on
the
other
hand,
has
few
alternatives
to
adequately
estimate
the
number
of
prime
ideals
with
a
prescribed
splitting
behaviour
in
a
Galois
extension
of
number
fields.
Unfortunately,
these
alternatives
are
not
sufficiently
robust
for
many
interesting
applications.
We
will
discuss
the
existing
literature
and
report
on
a
new
field-uniform
generalization
of
Brun-Titchmarsh
associated
to
the
Chebotarev
density
theorem.
This
result
has
consequences
for
counting
primes
represented
by
certain
binary
quadratic
forms
and
refining
unconditional
bounds
towards
the
Lang-Trotter
conjectures
for
elliptic
curves.
This
talk
is
based
on
joint
work
with
Jesse
Thorner.
M3 3103