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July 25th, 2013, 03:24 AM   #1
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Dirichlet characters

Hi! So i have made a table of all the nonprincipal Dirichlet characters mod 16. Is the following table correct?

n 1 3 5 7 9 11 13 15
X1(n) 1 1 1 1 1 1 1 1
X2(n) 1 i -i -1 -1 -i i 1
X3(n) 1 -1 -1 1 1 -1 -1 1
X4(n) 1 -i i -1 -1 i -i 1
X5(n) 1 1 -1 -1 1 1 -1 -1
X6(n) 1 i i 1 -1 -i -i -1
X7(n) 1 -1 1 -1 1 -1 1 -1
X8(n) 1 -i -i 1 -1 i i -1

I now want to verify from this table that for each real-valued nonprincipal character X mod 16, that: L(1,X) does not = 0. Any ideas on how to do this?

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July 25th, 2013, 04:28 AM   #2
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Re: Dirichlet characters

Quote:
Originally Posted by MrBibbles
Is the following table correct?
It seems so, yes.

Quote:
Originally Posted by MrBibbles
I now want to verify from this table that for each real-valued nonprincipal character X mod 16, that: L(1,X) does not = 0.
I am clearly not an expert on these things, all you can expect from me is the whole proof of Siegel that L(1, X) never vanishes

PS : All I have been able to do with elementary analysis that X1, X5 and X7 doesn't work.
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July 25th, 2013, 05:20 AM   #3
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Re: Dirichlet characters

Yeah. I can kinda see that. Iv been trying to research how to tackle this for ages, but their isnt very much information out there on dirichlet characters. It looks as though it should be some simple summation but I cant see what exactly i have to summate
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July 25th, 2013, 05:57 AM   #4
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Re: Dirichlet characters

Quote:
Originally Posted by MrBibbles
Hi! So i have made a table of all the nonprincipal Dirichlet characters mod 16. Is the following table correct?
At a glance, you listed all 8 Dirichlet characters mod 16. You didn't exclude the principal character.
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July 25th, 2013, 12:07 PM   #5
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Deleted. A dangerous flaw in the calculation.
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