What is this product?

[Max A.]

It is positive. Taking natural logarithm, we have
SUM(prime p; log((p-1)/p)/p)
It is easy to prove that if x<=1/2 then ln(1-x) >= -2x, implying
log((p-1)/p) >= -2/p.
Therefore,
SUM(prime p; log((p-1)/p)/p) >= -2*SUM(prime p; 1/p^2) = -2*P(2)
where P(k) is the prime zeta function. The value of P(2) is given by A085548.

Finally, we have
PROD(prime p; ((p-1)/p)^(1/p)) >= exp(-2*P(2)) > 0.4

Max

On 12/4/06, David Wilson <davidwwilson at comcast.net> wrote:
>
>
> What is PROD(prime p; ((p-1)/p)^(1/p))?  Is it 0 or positive?
>
> It might be interpreted as the expected value of phi(n)/n for very large n.