# [seqfan] Re: more digits of sum over squared inverse twin primes

Martin Fuller martin_n_fuller at btinternet.com
Thu Jun 4 10:40:55 CEST 2009

```Assuming the twin primes conjecture, we can estimate the contribution from the remaining twin primes as:
integral x^-2 * 4*C2 / log(x)^2 from 10^k to infinity

The estimates from 10^9 and 10^10 both give 0.237251776576663... although the last digit looks unreliable.

PARI:
\p60
C2=0.660161815846869573927812110014555778432623360284733413319448;
s=
{
[0.2115192743764172335600907029478458049886621315192743764172335601,
0.2362385924524670829013613789754682069943385757467958344217207676, 0.2372055599710870423370176800457876035334200633564086869752709814, 0.2372492660182123262653761802065117005961398895904169885978597094, 0.2372516057897405689878427120429396865698609342590217726341897254, 0.2372517642213452514702629520604252299216348196823717517418485243, 0.2372517756682939981077218099169625939398841251892729771263012051, 0.2372517765061919663119729774028233698105539917995612519644924169, 0.2372517765710358159350110748777318158842188919259177572926430824, 0.2372517765762034510704083224746640344084016190159084662884267692]
};
t=vector(#s, k, intnum(x=10^k, [1], x^-2 * 4*C2/log(x)^2))
u=s+t;
Mat(u)~

--- On Wed, 3/6/09, Robert G. Wilson, v <rgwv at rgwv.com> wrote:

> From: Robert G. Wilson, v <rgwv at rgwv.com>
> Subject: [seqfan] Re: more digits of sum over squared inverse twin primes
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Date: Wednesday, 3 June, 2009, 6:14 PM
> Seq. Fans,
>
>     Here are the partial sums to 64 decimals
> up to 10^k, k is the first
> number in parenthesis and the sum is the second.
>
>
> { 1,
> 0.2115192743764172335600907029478458049886621315192743764172335601}
> { 2,
> 0.2362385924524670829013613789754682069943385757467958344217207676}
> { 3,
> 0.2372055599710870423370176800457876035334200633564086869752709814}
> { 4,
> 0.2372492660182123262653761802065117005961398895904169885978597094}
> { 5,
> 0.2372516057897405689878427120429396865698609342590217726341897254}
> { 6,
> 0.2372517642213452514702629520604252299216348196823717517418485243}
> { 7,
> 0.2372517756682939981077218099169625939398841251892729771263012051}
> { 8,
> 0.2372517765061919663119729774028233698105539917995612519644924169}
> { 9,
> 0.2372517765710358159350110748777318158842188919259177572926430824}
> {10,
> 0.2372517765762034510704083224746640344084016190159084662884267692}
>
> Bob.
>
>
>
> Hagen von Eitzen wrote:
> >>>/ Richard Mathar
> >>
> >>
> >>
> >>So the constant c is less than
> >>    0.237251776574747 + lim(sum(1/k^2,{k,
> prime(130,000,001), n}, n ->
> >>infinity)
> >>  < 0.237251776574747 + 3.72376*10^(-10)
> < 0.237251776947124
> >>
> >>
> >
> >
> > The error term can be cut down to about 1/3 its size
> by summing only
> > over k= +-1 mod 6
> > and to about 1/5 its size by summing only over the
> relevant residues
> > -1,1, 11,13, 17,19 mod 30.
> > This should make c < 0.23725177665, I think;
> better, but unfortunately
> > no additional digit ...
> >
> > Hagen
> >
> >
> >>Hence 0.237251776574746 < c <
> 0.237251776947124 and we conclude that the
> >>first nine terms of the sequence are:  2, 3,
> 7, 2, 5, 1, 7, 7, 6
> >>
> >>Rigards,
> >>Farideh
>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>

```