- --- In primenumbers@yahoogroups.com,

"bhelmes_1" <bhelmes@...> wrote:

> My investigation confirms the Mertens probability

Thanks, Bernhard, for taking the time to check that conjecture.

Previously, Mike Oakes had done wonderful work, confirming it

to much better statistical accuracy.

Regarding your very limited statistics, at 50k digits,

I remark that> 50000 6 min < 115549 1:5200 1: 4631

is indeed in good accord with Mertens + PNT.

You found only 11 primes and it is pleasing that the ratio

5200/4631 differs from unity by less than 1/sqrt(11).

Alles gute, alles schöne!

David - --- In primenumbers@yahoogroups.com,

"bhelmes_1" <bhelmes@...> wrote:

> the distribution of primes

Since 1923, we have a had a very precise

> concerning the polynom f(x)=x^2+1

conjecture for the asymptotic density

of primes of the form x^2+1. See Shanks' review

http://www.ams.org/journals/mcom/1960-14-072/S0025-5718-1960-0120203-6/S0025-5718-1960-0120203-6.pdf

of the classic paper by G.H. Hardy and J.E. Littlewood:

"Some problems of 'Partitio numerorum'; III",

Acta Math. 44 (1923) pages 170.

The relevant Hardy-Littlewood constant,

1.3728134... is given, to 9 significant figures,

in Eq(3) of Shanks' paper.

More digits are easily obtainable from the methods in

"High precision computation of Hardy-Littlewood constants"

by Henri Cohen, available as a .dvi file from

http://www.math.u-bordeaux1.fr/~hecohen/

David