[seqfan] An attempt to generalize Mertens's third theorem

[Tomasz Ordowski]

Dear Reader!

Let F_{n}(x) = (Sum_{k<n} 1/k^x) (Product_{prime p<=n} (1-1/p^x)).

F_{n}(1) = 0, 1/2, 1/2, 11/18, 5/9, 137/225, 14/25, ... for n = 1, 2, ...

It seems that, by Mertens' third theorem,
Limit_{n->oo} F_{n}(1) = exp(-gamma) = 0.561...
where gamma = 0.577... is Euler's constant.

Cf. https://en.wikipedia.org/wiki/Mertens%27_theorems and
https://en.wikipedia.org/wiki/Euler%27s_constant#Generalizations

Let F(x) = Limit_{n->oo} F_{n}(x), for x > 0.
But can I assume that lim inf = lim sup ?
If so, find the value of F(1/2).
Is F(1) = exp(-gamma) ?
For x > 1, F(x) = 1.

Consider the function F(s) of the complex variable s,
without the analytic continuation of the zeta function,
in the critical strip 0 < Re(s) < 1. Is this area virgin?

Best regards,

Thomas Ordowski