## University COVID-19 update

### Questions about buildings and services? Visit the list of Modified Services.

Please note: The University of Waterloo is closed for all events until further notice.

# Number Theory Seminar Tuesday, October 8, 2019 — 1:30 PM EDT

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

### July 2021

S M T W T F S
27
28
29
30
1
2
3
4
6
8
9
10
11
13
15
16
17
18
20
22
23
24
25
27
30
31
1. 2021 (71)
1. August (4)
2. July (17)
3. June (15)
4. May (1)
5. April (4)
6. March (11)
7. February (9)
8. January (10)
2. 2020 (103)
1. December (10)
2. November (12)
3. October (4)
4. September (3)
5. August (1)
6. July (5)
7. June (1)
8. May (3)
9. March (16)
10. February (26)
11. January (22)
3. 2019 (199)
4. 2018 (212)
5. 2017 (281)
6. 2016 (335)
7. 2015 (211)
8. 2014 (235)
9. 2013 (251)
10. 2012 (135)