All Questions
Tagged with analytic-number-theory l-functions
154 questions
67
votes
17
answers
12k
views
Shortest/Most elegant proof for $L(1,\chi)\neq 0$
Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$?
Background: The non-vanishing ...
- 7,127
19
votes
1
answer
1k
views
constants in Gamma factors in functional equation for zeta functions.
Usually the Riemann zeta function $\zeta(s)$ gets multiplied by a "gamma factor" to give a function $\xi(s)$ satisfying a functional equation $\xi(s)=\xi(1-s)$. If I changed this gamma factor by a non-...
- 41.4k
38
votes
4
answers
6k
views
Modular forms and the Riemann Hypothesis
Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions?
What I'm thinking of is this: under the Mellin transform, the Riemann zeta function $...
- 889
19
votes
2
answers
2k
views
Applications of Artin's holomorphy conjecture
I wonder why the Artin conjecture is so important. The only reason I could figure out is that one could use the holomorphy of Artin L-series and Weil's converse theorem to show modularity of two-...
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