Inequality $p_n<n\left(\log n + \log\log n\right)$

Question

Wikipedia told me this bound (which of course holds true if $n\ge6$) for the $n$-th prime is due to Bach and Shallit, and I found "Eric Bach and Jeffrey Shallit: Algorithmic Number Theory, Volume I: Efficient Algorithms, MIT Press, August 1996.", can you confirm? If that's correct, what theorem is it? I went through what Google allowed me to read, but couldn't find it.

Answer 1

The book you mention appears to be a textbook; I do not think there is an implication that the result is due to Bach and Shallit, merely that it (possibly) can be found there.

Anyway, a better bound $p_n<n\left(\log n + \log\log n - 1/2 \right)$ was already established by Rosser and Schoenfeld in 1962.

You can find a nice overview on progress on bounding the $n$-th prime on a page of the TME-ETM project.

Answer 2

In your mentioned book, chapter 8.8: Some Useful Explicit Estimates, page 233, the result is given as Theorem 8.8.4 without a proof. As a reference, Rosser and Schoenfeld [1962] is given. In the bibliography, you can find

[Rosser and Schoenfeld 1962]: J.B. Rosser and L. Schoenfeld. Approximate formulas for some functions of prime numbers. Ill. J. Math. 6 (1962), 64-94.