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

[Hagen von Eitzen]

>
> >/ Richard Mathar
> />/ snapshots of results:
> />/ prime(900000)  0.23725177594...
> />/ prime(1290000) 0.23725177616...
> />/ ...
> />/ might test whether the 0.23725177 may turn to 0.23725178 later on.
> /
>
> No, 0.23725177 may not turn to 0.23725178.
> Because,
>
> prime(100,000,000)     0.237251776574074...
> prime(110,000,000)     0.237251776574342...
> prime(120,000,000)     0.237251776574562...
> prime(130,000,000)     0.237251776574746...
>
> 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
>
>