Lower bound for second Chebyshev function

Question

I was wondering is there any simple way to find nice lower bound for second Chebyshev function given by formula:

$$\psi(x)=\sum_{p\le x} \left\lfloor \frac{\ln x}{\ln p} \right\rfloor\ln p$$

that is inequality without any asymptotic notation?

Answer 1

Equations (3.37) and (3.14) of Rosser and Schoenfeld yield $$ x\left( 1-\frac{1}{2\log x} \right) +0.98x^{1/2} < \psi(x) $$ for $x \ge 563$.

That paper has other inequalities involving $\psi(x)$.

Added: A numerical check shows this inequality is actually valid for $x \ge 229$.