Rate of divergence of some integrals (Q1916618)

Let \[ M(x)= \sum_{n\leq x} \mu (n), \qquad \Psi (x)= \sum_{n\leq x} \Lambda (n), \qquad B(x)= \sum_{mn\leq x} {\textstyle {1\over m}} \mu(m), \] where \(\mu (n)\) is the Möbius function and \(\Lambda (n)\) the Mangoldt function. The authors state a general Tauberian theorem and derive from it \[ \int^x_1 (\Psi (t) -t)^2 t^{-2} dt\geq c_1\log x, \quad \int^x_1 M^2 (t) t^{-2} dt\geq c_2 \log x, \quad \int^x_1 B^2 (t) t^{-2} dt\geq c_3 \log x. \]