Thursday, October 5, 2017 1:30 pm
-
1:30 pm
EDT (GMT -04:00)
Lucile
Devin,
University
of
Ottawa
"Generalizations
of
Chebyshev's
bias"
Following
ideas
of
Rubinstein,
Sarnak
and
Fiorilli,
we
give
a general
framework
for
the
study
of
prime
number
races
and Chebyshev's
bias
attached
to
general
$L$-functions $L(s)
=
\sum_{n\geq
1}\lambda_{f}(n)n^{-s}$
satisfying
natural analytic
hypotheses.
We
put
the
emphasis
on
weakening
the
required hypotheses
such
as
GRH
or
linear
independence
properties
of
zeros of
$L$-functions.
In
particular
we
establish
the
existence
of
the logarithmic
density
of
the
set $\lbrace
x\geq
2
:
\sum_{p\leq
x}
\lambda_{f}(p)
\geq
0
\rbrace$ conditionally
on
a
much
weaker
hypothesis
than
was
previous known.
We
include
applications
to
new
Chebyshev's
bias
phenomena that
were
beyond
the
reach
of
the
previously
known
cases.
MC
5501