The following question concerns Nagura's proof that there is a prime between x and $\frac{6}{5}x$ for x > 2103. The question does not depend on a precise definition of the functions involved. We have
$T(x) = \sum_{m=1}^{\infty}\psi(\frac{x}{m})$
where $\frac{x}{m}$ is ordinary division. The claim reads:
$T(x) -T(\frac{x}{2})-T(\frac{x}{3})-T(\frac{x}{5}) + T(\frac{x}{30})$
$ = \psi(x) + \psi(\frac{x}{7})+\psi(\frac{x}{11})+...+\psi(\frac{x}{29})+...(*)$
$- \psi(\frac{x}{6})-\psi(\frac{x}{10})-...-\psi(\frac{x}{30})...(*) \leq\psi(x)......... (1) $
(*) is the author's note that the denominators in the remaining terms repeat under congruence mod 30.
The claim may well be true as stated, but I wonder if it shouldn't read $2 T(\frac{x}{30})$. Otherwise, if we do the bookkeeping,
$T(x) = \psi(x)+\psi(\frac{x}{2})+\psi(\frac{x}{3})... $
$-T(\frac{x}{2})=\psi(\frac{x}{2}+\psi(\frac{x}{4})+\psi(\frac{x}{6})...$
$-T(\frac{x}{3})=\psi(\frac{x}{3})+\psi(\frac{x}{6})+\psi(\frac{x}{9})... $
$-T(\frac{x}{5})=\psi(\frac{x}{5})+\psi(\frac{x}{10})+\psi(\frac{x}{15})... $
$+T(\frac{x}{30})=\psi(\frac{x}{30})+\psi(\frac{x}{60})+\psi(\frac{x}{90})... $
Each of the terms in the last line occur in all three of the previous (negative) sequences. For example, $\psi(\frac{x}{30})$ will occur in $T(\frac{x}{2}),T(\frac{x}{2}),T(\frac{x}{2})$. As written, it appears we should have $-2\psi(\frac{x}{30})$ in (1).
So is this likely a typo, or is the equation true as written? It seems to me the argument depends on keeping the denominators in the negative terms of (1) smaller than those in the positive terms.
I tried to preserve the question while keeping details to a minimum. More details on request, thanks for any suggestions.