Constants

[Tomasz Ordowski]

Hello Everyone!

Let R = Tanh(Integral_{x>=1} ArTanh((1 - T(x)) / (1 + T(x)))dx) = 0.930679... 

where T(x) = Zeta(2x) / Zeta(x)^2. The constant R is related to 

https://oeis.org/A348829 / https://oeis.org/A348830  

Whereas S = Integral_{x>=1} PrimeZeta(x)dx = 1.6366... 

Cf. https://oeis.org/A137245  (equality holds). 

More generally, for a constant c >= 1, 

S_c = Sum_{p prime} 1/(p^c log p) = 

= Integral_{x>=c} PrimeZeta(x)dx. 

Cf. https://oeis.org/A221711 

For what c, S_c = R ?  

Best, 

Tom Ordo 

____________

https://oeis.org/draft?user=Thomas%20Ordowski