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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 V (p. 3). Let $f$ be additive, and suppose there are constants $c_1,c_2$ and infinitely many $n$ for each of which there are integers $a_1<a_2<\cdots<a_x\le n$ with $x>c_1n$ and $|f(a_i)-f(a_j)|<c_2$ for all $i,j$. Then there is a constant $c$ such that, with $f^+(p)=f(p)-c\log p$ and $(f^+)'$ its truncation,

$$\sum_p\frac{((f^+)'(p))^2}{p}<\infty.$$

The print writes the series as $\sum_p((f^+)'(p)/p)^2$, which converges for every $c$ because $|(f^+)'(p)|\le1$; the paper's converse (p. 3), its proof (which ends on p. 14 with $\sum(\varphi'(p))^2/p<\infty$) and its use in Theorems IV and XI all take the form above.

Converse (p. 3). If $f(p)=c\log p+f^+(p)$ with $\sum_p((f^+)'(p))^2/p<\infty$, then for every $c_1<1$ there is a $c_2$ such that for every $n$ there are $a_1<\cdots<a_x\le n$ with $x>c_1n$ and $|f(a_i)-f(a_j)|<c_2$.

Definition (p. 3). $f$ is finitely distributed when it satisfies the hypothesis of Theorem V.

Proof pointer

The converse is proved first (pp. 8--10), by a second-moment estimate for $f^+(m)$ around $A_n=\sum_{p\le n}f^+(p)/p$: for every $c_2<1$ there is a $c_4$ with $|f(m)-c\log n-A_n|<c_4$ for more than $c_2n$ integers $m\le n$. The theorem itself (pp. 10--14) splits into three cases on pairs of prime sequences $p_i,q_i$ with $p_i/q_i\to c\in(1,\infty)$ and $\sum1/p_i=\sum1/q_i=\infty$: $f(p_i)-f(q_i)\to\pm\infty$ (Case 1), $f(p_i)-f(q_i)\to0$ with $\sum(f'(p))^2/p=\infty$ (Case 2), and $f(p_i)-f(q_i)\to d$, printed with $1<d<\infty$ (Case 3, reduced to Case 2 by subtracting a multiple of $\log m$); in each the finitely distributed hypothesis is contradicted, Case 2 following an earlier density lemma of Erdős.

Read depth

Claims checked: the statement, the converse and the definition read on the page image of p. 3; the proof on pp. 8--14 read for structure. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: Turán's method and Lemma 2 of Erdős's 1937 paper on the density of some sequences of numbers.

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

• Problem 491: the paper says (p. 3) that it deduces from Theorem V the results proved as Theorems XI and XIII, which give $f(n)=c\log n$ for nondecreasing $f$ and for $f$ with $f(n+1)-f(n)\to0$; the proof of Theorem XI (pp. 17--18) uses Theorem V, while the written proof of Theorem XIII (pp. 18--19) does not. Theorem V itself does not address the problem's bounded-difference hypothesis.