# Bookkeeping question in claim about arithmetic functions in a proof

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.

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What is T(x)? I haven't heard about a number theory function that is often called T(x). –  Mayank Pandey 2 days ago
@MayankPandey: I haven't looked at this in 2 years but T(x) is defined in the first formula: a sum of $\psi$ functions. It occurs in Nagura's 1952 paper. –  daniel 2 days ago
Oh. I thought T(x) was something different that happened to satisfy the first statement. –  Mayank Pandey yesterday

Perhaps I'm missing something obvious here, but doesn't the $\psi(\frac{x}{30})$ term appear once in each of $T(x),T(\frac{x}{2}),T(\frac{x}{3}),T(\frac{x}{5}),T(\frac{x}{30})$ meaning that it is added twice and subtracted three times in (1) giving the equation as written by the author?

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No, you are right! I forgot it would occur in the first line. Thanks. –  daniel Dec 24 '11 at 4:19