On the logarithmic derivative of the Euler product

$$ prod_p(1 - ((1) / (p^s)))^-1 := ζ(s) , (Re(s) > 1). $$

Some of the results deal with extremal bounds for derivatives of the Riemann zeta function ζ(s), and have a potential to have applications in number theory and cryptography, especially when combined with other analytic and algebraic results.