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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.

Theorem I (p. 1). Let $f$ be real additive with $f(p)\to0$ as $p\to\infty$ and $\sum_p (f'(p))^2/p=\infty$, and put $F(m)=f(m)-[f(m)]$, with $[a]$ the greatest integer $\le a$. Then the distribution function of $F(m)$ is $x$: for each $c$ the integers $m$ with $F(m)\le c$ have density $c$.

The paper notes (p. 1) that $f(p)\to0\pmod 1$ suffices, and (p. 2) that without such a hypothesis the conclusion can fail: $f(p)=\tfrac12$, $f(p^\alpha)=0$ for $\alpha>1$ gives the distribution function $\tfrac12$ on $[0,\tfrac12]$ and $1$ on $[\tfrac12,1]$.

Proof pointer

Pp. 5--8. Lemma 1 (p. 6) is a Berry-type error bound for the normal approximation to the density of the truncated sums $f_{u,v}(m)=\sum_{u\le p\le v,\,p\mid m}f(p)$; Lemmas 2 and 3 show that these sums are uniformly distributed mod 1 in density; Lemmas 4--6 carry this to the counts up to $n$ by the method of the Erdős--Kac paper; Lemma 7 (p. 8) shows $f$ and $f_{1,v}$ differ by more than $\epsilon$ only on few $m\le n$, which gives the theorem.

Read depth

Claims checked: the statement and hypotheses read on the page images of the print; the proof on pp. 5--8 read for structure. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: Berry's theorem and the Erdős--Kac paper (the paper's reference I).

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.