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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 III (p. 2). Let $f$ be additive, and suppose that for some constant $c$ the function $\psi(m)=f(m)-c\log m$ satisfies the hypotheses of Theorem II. Then the paper writes

$$\varphi(m)=f(m)-c\log m-\sum_p\frac{\psi'(p)}{p} =f(m)-\sum_{p\le n}\frac{f(p)}{p}+c+o(1)$$

and asserts that $\varphi(m)$ has a distribution function. Continuity and strict increase are not asserted here.

The paper calls Theorem III essentially identical with Theorem II, and says (p. 2) that the converse is probably true: if $f(m)-\sum_p f(p)/p$ has a distribution function then $f(m)=c\log m+\varphi(m)$ with $\sum_p(\varphi'(p))^2/p<\infty$; it can prove this converse only when $f(p)>0$.

Proof pointer

No separate proof is given; the paper presents it as a slightly stronger form of Theorem II, whose proof it omits.

Read depth

Claims checked: the statement read on the page image of p. 2. Nothing here is independently reviewed.

Dependencies

Theorem II.

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.