Where did the constant with $\vartheta(x)<1.01624x$

Question

In:

Rosser, J. Barkley; Schoenfeld, Lowell (1962). "Approximate formulas for some functions of prime numbers.". Illinois J. Math. 6: 64–94.

Theorem 9.

$\vartheta(x)< 1.01624 x$ for $0 < x$.

Where did this constant of 1.01624 come from? Can it be improved?

By improved, I mean like this one for Theorem 12.

"The quotient $\psi(x)/x$ takes its maximum at $x = 113$, and $\psi(x) < 1.03883 x$ for $0 < x$."

"In precise terms this constant is log(955888052326228459513511038256280353796626534577600)/113"

"1.0388205776091298930081555627382465269336112084545034825058980..."

both quotes from A206431 in the OEIS. https://oeis.org/A206431

Now that is concrete.

I did find that Dusart in 2010 proved in Proposition 5.1 that

$\vartheta(x) − x < \frac{1}{36260} x$ for $x > 0$.

I think this (because they are using tables to find the value) implies that there is not a concrete value like $\psi(x)$, correct?

Answer 1

see page 80, they say verified up to $10^8$ by theorem 18, then up to $10^{16}$ by theorem 24, then Table I with $\theta < \psi,$ which is automatic by the definitions. Table I is on page 90.

It cannot be improved very much regardless, as $\theta$ and $\psi$ are asymptotic to $x,$ that is the Prime Number Theorem.