# Nagura's paper--can we substitute for the original upper bound?

This question concerns two results about primes. The first is J. Nagura's 1952 result, that there is a prime on the interval $[x, (1+1/5)x]$ for $x> 2103,$ which depends on the result derived there: $\psi(x) < 1.086x,$ (Lemma 2, p. 179), here., and the second is Rosser's 1962 result: $\psi(x) < 1.03883x,$ (Theorem 12, p. 71), here.

My question is whether we can simply insert Rosser's improvement for the upper bound in the last expression in Nagura's paper, an expression which is derived on the last page independent of previous results, to obtain a slightly better bound?

If we do this, the expression Nagura uses to establish a lower bound for $\vartheta(\frac{n+1}{n}x)-\vartheta(x)$ on the last page of his paper becomes positive around x = 3225 for n = 7. So we have the slight improvement, that there is always a prime on $(x, (1 + 1/7) x.)$

Maybe this is somewhat obvious. Or wrong. I do see that by 1962 there was no mathematical importance in a marginal improvement over Nagura's result, I am just trying to sort out how some of these older results relate to each other.

The question concerns the last equation in Nagura on page 181 before the results, in the middle of the page:

$$\vartheta(\frac{n+1}{n}x) - \vartheta(x) \geq 0.916(\frac{n+1}{n}x +\sqrt{x}+ \sqrt[3]{x})- 6.954 - 1.086(x + \sqrt{\frac{x(n+1)}{n}}+ \sqrt[3]{\frac{x(n+1)}{n}}+\sqrt[5]{\frac{x(n+1)}{n}}).$$

[The constant 6.954 is associated with the lower bound 0.916. ]

Thanks for any insight.

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The reason I was not sure is that I didn't find a result between 1952 and 1962 improving on Nagura. Having glanced at Rosser and Schoenfeld's series of papers I see now that not only were they confident of a different order of results, but Rosser had the result $\psi(x)<1.038821x$ in his 1941 paper. Rosser, Explicit Bounds for some Functions on Prime Numbers, p.228, Am. J. Math., 63(1), (1941). This does not detract from Nagura's result but explains why it does not figure in later papers.