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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 $(m_1,m_2)=1$. The truncation $f'$ is $f'(p)=f(p)$ when $|f(p)|\le1$ and $f'(p)=1$ otherwise.

Theorem IV (p. 2, quoted). "Let $f(m)$ be an additive function such that $\sum_{f(p)\ne0}1/p$ diverges. Then to every $\epsilon$ there exists a $\delta$ such that if $a_1<a_2<\cdots<a_x\le n$ is a sequence of integers with $|f(a_i)-f(a_j)|<\epsilon$ then $x<\delta n$ [sic] for $n$ sufficiently large."

As printed the statement is trivial ($\delta=2$ always works). The proof (pp. 14--17) establishes it with the roles of the constants exchanged: for every $\epsilon>0$ there is a $\delta>0$ such that, for $n$ sufficiently large, any integers $a_1<\cdots<a_x\le n$ whose values $f(a_i)$ all lie in an interval $(D,D+\delta)$ number $x<\epsilon n$ (display (12), p. 14, and the closing contradiction on p. 17).

The paper paraphrases the theorem (p. 17): if $\sum_{f(p)\ne0}1/p=\infty$, the distribution function tries to be continuous whether it exists or not.

Proof pointer

Pp. 14--17. Theorem V first makes $f$ finitely distributed, so $f(m)=c\log m+\varphi(m)$ with $\sum(\varphi'(p))^2/p<\infty$. For $c=0$ the proof uses Lemmas 8 and 9 (pp. 14--15): $f(m)$ is close to its truncation $f_k(m)$ plus a constant for most $m$, and $f_k$ rarely lands in a short interval. For $c\ne0$ a lemma on pairs $a_i/p_i=a_j/p_j$ (pp. 15--16) produces two members whose $f$-values differ by more than $c\log(1+c_1)-2\eta$, a contradiction.

Read depth

Claims checked: the statement read on the page image of p. 2 and the quantifier order checked against the proof on pp. 14--17, which was read for structure. Nothing here is independently reviewed.

Dependencies

Theorem V. External inputs named by the paper: Turán's method and Erdős's paper "On the density of some sequences of numbers III" (1938).

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.