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Statement

Setting (p. 1). $f$ is a real additive function: $f(m_1m_2)=f(m_1)+f(m_2)$ whenever $(m_1,m_2)=1$. A function $\psi$ is the distribution function of $f$ when $\psi(-\infty)=0$, $\psi(\infty)=1$ and, for every real $c$, $\psi(c)=\lim_{n\to\infty}N(f;c,n)/n$, where $N(f;c,n)$ counts the $m\le n$ with $f(m)\le c$. The truncation $f'$ is $f'(p)=f(p)$ when $|f(p)|\le1$ and $f'(p)=1$ otherwise.

Hypotheses (p. 2): $\sum_p (f'(p))^2/p<\infty$ and $\sum_p f'(p)/p$ diverges.

Theorem II (p. 2, quoted). "Put $\varphi(m)=f(m)-\sum_p \frac{f'(p)}{p}$ [sic]. Then $f(m)$ [sic] has a distribution function, and the distribution function is continuous and strictly increasing in $(-\infty,+\infty)$."

As printed the statement does not parse: the sum over all $p$ diverges by hypothesis, and by the Wintner--Erdős criterion recalled on p. 1 $f(m)$ itself has no distribution function under these hypotheses. The intended reading is that $\varphi(m)$, with the sum truncated, has the distribution function; Theorem III (p. 2) writes the centring in the form $\sum_{p\le n}$, with $n$ the range $m\le n$ over which densities are taken.

Proof pointer

Not proved in the paper: the proof is omitted as similar to one in an earlier paper of Erdős (p. 2). The remark on p. 15 says the proof of Theorem IV for $c=0$ shows the continuity, which it calls the hardest part of Theorem II.

Read depth

Claims checked: the statement read on the page image of p. 2. No proof is given in the paper. Nothing here is independently reviewed.

Dependencies

None in the corpus. The paper relies on the Wintner--Erdős criterion for the existence of a distribution function (p. 1).

Source. P. Erdős, On the distribution function of additive functions, Ann. of Math. (2) 47 (1946), 1--20, doi:10.2307/1969031; the edition read is named on the source card.

Bears on

None directly.