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Titanic Golomb-Dickman prime 

djbroadhurst     Message 45 of 49  Apr 2, 2010
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The Golomb-Dickman constant gives the average value of
log(p)/log(n), where p is the largest prime divisor of n.

For background, see
http://mathworld.wolfram.com/Golomb-DickmanConstant.html
http://mathworld.wolfram.com/DickmanFunction.html
though be advised that in several places Eric
writes an argument 1/(1-t) when he really means t/(1-t).

It took Pari-GP less than an hour to compute 1700 digits of the
Golomb-Dickman constant. The first 1659 digits form the prime
http://physics.open.ac.uk/~dbroadhu/cert/p1659.in
with an ECPP proof of primality in
http://physics.open.ac.uk/~dbroadhu/cert/p1659.out
and a Pari-GP evaluation of the residuum as follows:

default(realprecision,1700);
lambda=intnum(x=0,1,exp(-eint1(-log(x))));
p1659=readvec("p1659.in")[2];
print(precision(lambda-p1659/10^1659,1));

3.788386996723134801 E-1660

Perhaps a Mma enthusiast might like to check this residuum?

David (from a Cohen-rich zone)
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djbroadhurst     Message 46 of 49  Apr 2, 2010
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--- In primeform@yahoogroups.com, 
"djbroadhurst" <d.broadhurst@...> wrote:

> It took Pari-GP less than an hour to compute 1700 digits of the
> Golomb-Dickman constant. 
> The first 1659 digits form the prime
> http://physics.open.ac.uk/~dbroadhu/cert/p1659.in

Eric has sent me an impressively fast check by Mma:

NIntegrate[\[Lambda] = Exp[LogIntegral[x]], {x, 0, 1},
WorkingPrecision -> 1700, Method -> "DoubleExponential"] // Timing
{328.749, \

confirming the digits.

David
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djbroadhurst     Message 47 of 49  Apr 3, 2010
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--- In primeform@yahoogroups.com, 
"djbroadhurst" <d.broadhurst@...> wrote:

> For background, see
> http://mathworld.wolfram.com/Golomb-DickmanConstant.html
> http://mathworld.wolfram.com/DickmanFunction.html
> though be advised that in several places Eric
> writes an argument 1/(1-t) when he really means t/(1-t).

Eric has now corrected those 3 typos and has added 
a larger titanic Golomb-Dickman prime to
http://mathworld.wolfram.com/Golomb-DickmanConstant.html

David
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djbroadhurst     Message 48 of 49  Apr 6, 2010
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In primeform@yahoogroups.com, 
"djbroadhurst" <d.broadhurst@> wrote:

> 200 digits for the asymptotic probability that a large number N has 
> it greatest prime factor less than N^alpha, in the rational cases 
> with 1/alpha = 2,3,4,5,6,7,8.

I have updated
http://physics.open.ac.uk/~dbroadhu/cert/smoctic.txt
with 800 good digits for the Dickman probabilities with 
1/alpha = 2,3,4,5,6,7,8
and (after some harder work) with 400 good digits for
1/alpha = 9,10,
thanks to two theorems proven in
http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.0519v1.pdf
which acknowledges Mike Oakes' grand support, via this list.

David Broadhurst
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mikeoakes2     Message 49 of 49  Apr 9, 2010
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--- In primeform@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
>
> I have updated
> http://physics.open.ac.uk/~dbroadhu/cert/smoctic.txt
> with 800 good digits for the Dickman probabilities with 
> 1/alpha = 2,3,4,5,6,7,8
> and (after some harder work) with 400 good digits for
> 1/alpha = 9,10,
> thanks to two theorems proven in
> http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.0519v1.pdf
> which acknowledges Mike Oakes' grand support, via this list.

That's an amazing chunk of work, David.
(All I really did was to initiate what turned out to be a promising path from quadratic to cubic to ... smoothness, then hang on to your coattails as you took off into the Empyrean:-)

A most astonishing result is your (conjectured) generating function (15).
It seems to me there's the usual giant, silent, Broadhurst-type step from (14) to (15) - can you maybe fill in some details of how to get there?

Mike
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