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Source: published paper, printed p. 414 (PDF p. 4), Section 4. The paper gives the following further assertions without proof; this page does not upgrade them to complete proof claims.

With $\theta(x)=\sum_{p\le x}\log p$ as on p. 412, the source states

$$|\theta(x)-x|\le0.006788\,\frac{x}{\log x} \qquad(x\ge2.89\cdot10^7),$$ $$\begin{aligned} p_k&\le k(\log k+\log\log k-0.9484)&& (k\ge39017),\\ p_k&\le k\left(\log k+\log\log k-1+ \frac{\log\log k-1.8}{\log k}\right)&& (k\ge27076),\\ p_k&\ge k\left(\log k+\log\log k-1+ \frac{\log\log k-2.25}{\log k}\right)&& (k\ge2). \end{aligned}$$

It further states that these bounds show that for $x\ge3275$ the interval

$$\left[x,x+\frac{x}{2\log^2x}\right]$$

contains at least one prime. Defining $\pi(x)$, "as usual", as "the number of primes lower than $x$", it gives

$$\pi(x)\ge\frac{x}{\log x}\left(1+\frac{0.992}{\log x}\right) \quad(x\ge599),$$ $$\pi(x)\le\frac{x}{\log x}\left(1+\frac{1.2762}{\log x}\right) \quad(x>1).$$

The printed definition counts primes strictly below $x$, while the usual convention counts primes $p\le x$; the two differ only when $x$ is prime, and the source does not say which it intends beyond calling its definition the usual one. The paper gives no proof of any statement on this page, and none of them is an input to Theorem 3 or to the compiled covering-system applications.