Question

SOME PROPERTIES OF THE SERIES OF COMPOSED NUMBERS, p2 gives the bounds

$$ l(x)=\frac{x}{\log(x)-28/29}<\pi(x) < u(x)=\frac{x}{\log(x)-1.12} \qquad (1)$$

for $x \geq 3299$

TWO GENERALIZATIONS OF LANDAU’S INEQUALITY, p6 gives

$$ x\pi(x) < 2 \pi(x^2) $$ for $x \geq 67$

To show $\pi(x^2) > x\pi(x)/2 > \pi((x-1)^2)$ tried the bounds in (1) leading to:

$$x l(x) /2 > u((x-1)^2) \iff $$ $$ \frac{29 \, x^{2}}{2 \, {\left(29 \, \log\left(x\right) - 28\right)}} - \frac{{\left(x - 1\right)}^{2}}{\log\left({\left(x - 1\right)}^{2}\right) - 1.12} >0 \qquad (2)$$

Experimentally the LHS of (2) is increasing up to $10^{120}$ and is quite large.

1. Is this reasoning correct?
2. Can (2) be proved?

All CAS I tried failed to solve (2) though the limit at infinity is $+\infty$.

Answer 1

Note that, in the second paper, the inequality $$x\pi(x)<2\pi(x^2)$$ is derived from the inequality $$xy\pi(x)\pi(y)>4\pi^2(xy)$$ by setting $x=y$. The proof for theorem 3 seems correct to me, so the sign flip is definitely a typo :)