Bill Allombert on Fri, 18 May 2012 18:23:14 +0200

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Re: zetap

On Fri, May 18, 2012 at 06:04:53PM +0200, sou nonyma wrote:
> i found that i was not clear so here are few value of my algorithme:
> in the formula:
> zetap(4)=3+2*3^2+2*3^6+2*3^7+O(3^8)       (i have change a few things since
> i put the previous message)
> <x>=<1/p>=1+O(3^8)
> the conbination C(1-s,1) for exemple C(-3,1)=
> 2+2*3+2*3^2+2*3^3+2*3^4+2*3^5+2*3^6+2*3^7+O(3^8)
> the bernoulli number 6 B(6)=2*3^-1+3^2+2*3^3+2*3^4+3^5+O(3^6)
> and the term for k=1  which is =2*3^3+3^4+3^5+3^6+3^7+O(3^8)

Well, since zeta(-5) is -1/252, by definition of the p-adic zeta function, we must have
zeta(4+O(3^2)) = zeta(-5+O(3^2)) = -1/252+O(3^-1) = 2*3^-2+O(3^-1)
which is what PARI gives.

Look at your formula to see why you do not get the right valuation.