Fan Ge, Pure Mathematics, University of Waterloo
"The number of zeros of $\zeta'(s)$
The
distribution
of
zeros
of
the
derivative
of
the
Riemann
zeta-function
is
closely
related
to
that
of
zeta
itself.
One
of
the
basic
questions
in
the
study
of
zeros
is
the
zero-counting
problem.
In
particular,
the
error
terms
in
the
zero-counting
formulas
are
of
special
interest.
For
the
Riemann
zeta-function
the
best
known
bound
for
the
error
term
is
O(log
T)
due
to
von
Mangoldt
in
1905.
If
we
assume
the
Riemann
Hypothesis
(RH),
then
the
essentially
best
bound
is
O(log
T/loglog
T)
due
to
Littlewood
in
1924.
For
the
derivative
of
the
Riemann
zeta-function
Berndt
proved
the
bound
O(log
T)
unconditionally
in
1970,
and
assuming
RH
Akatsuka
proved
O(log
T/sqrt(loglog
T))
in
2012.
We
show
that
on
RH,
the
error
term
in
the
zero-counting
formula
for
the
derivative
of
Riemann
zeta
is
O(log
T/loglog
T),
thus
it
is
of
the
same
size
as
that
for
zeta
itself.