Ertan Elma, Department of Pure Mathematics, University of Waterloo
"Discrete Mean Values of Dirichlet L-functions"
Let
χ
be
a
Dirichlet
character
modulo
a
prime
number
p
⩾
3
and
let
\mathfrak
a_χ:=(1-χ(-1))/2.
Define
the
mean
value
\begin{align*}
\mathcal{M}_{p}(s,\chi):=\frac{2}{p-1}\sum_{\substack{\psi
\bmod
p\\\psi(-1)=-1}}L(1,\psi)L(s,\chi\overline{\psi})
\end{align*}
for
a
complex
number
s
such
that
s≠
1
if
\mathfrak
a
_χ=1.
Mean
values
of
the
form
above
have
been
considered
by
several
authors
when
χ
is
the
principal
character
modulo
p
and
\Re(s)>0
where
one
can
make
use
of
the
series
representations
of
the
Dirichlet
L-functions
being
considered.
In
this
talk,
we
will
investigate
the
behaviour
of
the
mean
value
\mathcal{M}_{p}(-s,χ)
where
χ
is
a
nonprincipal
Dirichlet
character
modulo
p
and
\Re(s)>0.
Our
main
result
is
an
exact
formula
for
\mathcal{M}_{p}(-s,χ)
which,
in
particular,
shows
that
\begin{align*}
\mathcal{M}_{p}(-s,\chi)=
L(1-s,\chi)+\mathfrak
a_\chi
2p^sL(1,\chi)\zeta(-s)+o(1),
\quad
(p\rightarrow
\infty)
\end{align*}
for
fixed
0<σ:=\Re(s)<\frac{1}{2}
and
|\Im
s|=o\left(p^{\frac{1-2
σ}{3+2
σ}}\right).
MC 5417