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Source: published paper, printed pp. 412–415 (PDF pp. 2–5), the introduction, Theorem 3 and references. These are external inputs to the reconstructed proof; their full proofs and large computations are not reproduced here.

Finite zero verification

Let $N(T)$ be the number of zeros $\beta+i\gamma$ of $\zeta$ with $0<\gamma\le T$; the paper does not say whether zeros are counted with multiplicity. With $F$ and $A$ defined at Theorem 1, Dusart imports

$$N(A)=F(A)=1500000001,\qquad \beta=\tfrac12\text{ for every such zero with }0<\gamma\le A.$$

He cites R. P. Brent, J. van de Lune, H. J. J. te Riele and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. II, Mathematics of Computation 39 (1982), 681–688; and J. van de Lune, H. J. J. te Riele and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. IV, Mathematics of Computation 46 (1986), 667–681.

The separate finite certificate only proves an interval for the root of $F(A)=1500000001$. An interval for that root alone would not prove any claim about $N(A)$ or the real parts of zeros.

Robin's estimates

For $k\ge3$, the proof uses

$$\theta(p_k)\ge k\left(\log k+\log\log k-1+ \frac{\log\log k-2.1454}{\log k}\right), \tag{1}$$

where $\theta(x)=\sum_{p\le x}\log p$. Equation numbers on this page are its own: the print's (1) on p. 413 is this estimate, and its (2) is the final-range estimate (4) below. This is cited to p. 376 of G. Robin, Estimation de la fonction de Tchebychef $\theta$ sur le $k$-ième nombre premier et grandes valeurs de la fonction $\omega(n)$, nombre de diviseurs premiers de $n$, Acta Arithmetica 42 (1983), 367–389.

Dusart also invokes Lemma 3 on p. 375 of that source for the range

$$3\le p_k\le10^{11} \quad\Longrightarrow\quad p_k\ge k(\log k+\log\log k-1). \tag{2}$$

The range in (2) is stated in terms of the prime $p_k$, not the index $k$. This compilation does not enumerate the primes to $10^{11}$ or independently reconstruct Robin's proof of that range. Its weak inequality is retained; no all-$k$ strict conclusion is inferred from a finite check that was not run.

Schoenfeld's estimates

The paper states both bounds only at $x=p_k$. The first large-prime range uses the one-sided bound

$$\theta(p_k)-p_k\le0.0077629\,\frac{p_k}{\log p_k} \qquad(10^{11}\le p_k\le e^{500}). \tag{3}$$

The final range uses

$$|\theta(p_k)-p_k|\le 1.657\cdot10^7\,\frac{p_k}{\log^4p_k} \qquad(p_k\ge e^{1800}). \tag{4}$$

Dusart cites respectively pp. 357 and 360 of L. Schoenfeld, Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II, Mathematics of Computation 30 (1976), 337–360. The relevant older bounds and computations are external; they are not replaced by an assumed asymptotic prime number theorem.

Finally, Theorem 1 is the explicit analytic estimate imported from J. B. Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$, Mathematics of Computation 29 (1975), 243–269, Theorem 4. Its exact formula is written out there. The complete numerical specialization and three-range deduction are the same-paper work supplied here.