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Erschienen in:
2008 | OriginalPaper | Buchkapitel
Non-vanishing of Dirichlet L-functions at the Central Point
verfasst von : Sami Omar
Erschienen in: Algorithmic Number Theory
Verlag: Springer Berlin Heidelberg
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This paper deals with the matter of the non-vanishing of Dirichlet
L
-functions at the central point for all primitive characters
χ
. More precisely, S. Chowla conjectured that
$L(\frac{1}{2},\chi)\not =0$
, but this remains still unproved. We first give an efficient algorithm to compute the order
n
χ
of zero of
L
(
s
,
χ
) at
$s=\frac{1}{2}$
. This enables us to efficiently compute
n
χ
for
L
-functions with very large conductor near 10
16
. Then, we prove that
$L(\frac{1}{2},\chi)\not =0$
for all real characters
χ
of modulus less than 10
10
. Finally we give some estimates for
n
χ
and the lowest zero of
L
(
s
,
χ
) on the critical line in terms of the conductor
q
.