--- In

primenumbers@yahoogroups.com,

"djbroadhurst" <d.broadhurst@...> wrote:

> Provided that p << N, I guess that the average value

> of omega(n/p) for the numbers n up to N with least

> prime divisor p is

>

> log(log(N)) + B_1 - sum(prime q < p, 1/q) + o(1)

>

> with B_1 =

> 0.261497212847642783755426838608695859051566648261199206192...

I have proven this for primes p up to 19,

but the general proof eludes me.

My proof for p = 19 was already laborious and is

equivalent to showing that test(19) is true, here:

{T(n)=local(F=factor(n)[,1]);

sum(k=1,n,sum(j=1,#F,gcd(k,F[j])==1))/n^2;}

{test(p)=local(P=1,D=1,A=1-1/p);

forprime(q=2,p-1,P*=q;D*=1-1/q;A-=1/q);

sumdiv(P,n,moebius(n)*T(n*p))*p/D == A;}

{gettime;forprime(p=2,19,if(test(p),

print("p = "p" proven in "gettime" ms")));}

p = 2 proven in 0 ms

p = 3 proven in 0 ms

p = 5 proven in 0 ms

p = 7 proven in 0 ms

p = 11 proven in 8 ms

p = 13 proven in 141 ms

p = 17 proven in 2926 ms

p = 19 proven in 66089 ms

Challenge: prove that test(p) is true for all prime p.

David (unable to this, at present)