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

[Tomasz Ordowski]

P.S.  F(s) in the critical strip 0 < Re(s) < 1.

F(1/2) = 1/exp(zeta(1/2)+2) = 0.582954878...

F(s) = 1/exp(zeta(s)+1/(1-s)) for Re(s) > 1/2
if and only if the Riemann hypothesis is true.

T. Ordowski
________________
Note that lim_{s->1} zeta(s)+1/(1-s) = gamma,
so F(1) = 1/exp(gamma) = exp(-gamma), qed.

sob., 28 sty 2023 o 15:31 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> 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
>
>