[seqfan] Z(1) = exp(-gamma)

[Tomasz Ordowski]

Dear readers!

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

Z_{n}(1) = 1/1, 3/4, 11/18, 25/36,137/225, 49/75, 726/1225, ... for n =
1,2,...
The sequences of these numerators and denominators are not in the OEIS.

Z(1) = Limit_{n->oo} Z_{n}(1) = exp(-gamma) = 0.561...
where gamma = 0.577... is Euler's constant.

Cf. https://en.wikipedia.org/wiki/Euler%27s_constant#Exponential

Let Z(x) = Limit_{n->oo} Z_{n}(x) for x > 0.

Find the value of Z(1/2).
Z(1) = exp(-gamma).
For x > 1, Z(x) = 1.

Consider the function Z(s) of the complex variable s,
without the analytic continuation of the zeta function,
in the critical strip 0 < Re(s) < 1.

Good luck!

Thomas Ordowski