Erdos 1946 distribution function additive functions

../

conjecture_p3: Erdős's statement that an additive function with f(n + 1) - f(n) < c_1 for all n probably equals c log n plus a bounded function, and his conjectures that f(n) = c log n when f(n + 1) >= f(n), or when f(n + 1) - f(n) tends to 0, outside a set of density 0.

theorem_1: Erdős's theorem that when f(p) tends to 0 and the sum of f'(p)^2/p over primes diverges, the fractional part f(m) - [f(m)] of the additive function has the distribution function x on [0,1].

theorem_11: Erdős's theorem that an additive function with f(m + 1) >= f(m) for every m equals c log m for a constant c.

theorem_13: Erdős's theorem that an additive function whose consecutive differences f(m + 1) - f(m) tend to 0 equals c log m for a constant c.

theorem_2: Erdős's theorem that when the sum of f'(p)^2/p converges and the sum of f'(p)/p diverges, the additive function centred by the partial sums of f'(p)/p has a distribution function continuous and strictly increasing on the whole real line; the printed statement needs reading, as noted.

theorem_3: Erdős's theorem that if f(m) - c log m satisfies the hypotheses of Theorem II for some constant c, then f(m) minus the partial sum of f(p)/p over p <= n, plus c, has a distribution function.

theorem_4: Erdős's theorem that when the sum of 1/p over primes with f(p) != 0 diverges, for every eps > 0 there is a delta > 0 such that fewer than eps n integers up to n have f-values in one interval of length delta, for large n; the print states the quantifiers in a trivial order, as noted.

theorem_5: Erdős's structural theorem: if for infinitely many n more than c_1 n integers up to n have f-values pairwise within c_2, then for some constant c the truncation of f^+(p) = f(p) - c log p satisfies sum_p (f^+)'(p)^2/p < infinity; the printed series needs correcting, as noted. The paper proves the converse too.

theorem_6: Erdős's law of the iterated logarithm for an additive function with bounded prime values: the truncated sums over prime divisors of m exceed A_u + (1 + eps) sqrt(2 B_u log log B_u) for some u > d only on a set of upper density tending to 0, while the 1 - eps level is exceeded almost always.

---

P. Erdős: On the distribution function of additive functions, Ann. of Math. (2) 47 (1946), 1--20 MR 7,416c; Zentralblatt 61,79. The copy read for this card is the Rényi Institute's Erdős archive scan, which prints no notice; the article's JSTOR page for DOI 10.2307/1969031 could not be loaded on 2026-10-02 and its Crossref record names no license, and the journal's site shows the footer "Copyright © 2026 Annals of Mathematics" and names no license (https://annals.math.princeton.edu/, read 2026-10-02), every other right reserved.

Erdos studies limiting distribution functions of real additive arithmetic functions f, continuing his work with Wintner, whose criterion for the existence of a distribution function the paper recalls on p. 1, and with Kac. Theorem I (p. 1) shows that if f(p) tends to 0 and sum_p f'(p)^2/p diverges, the fractional part F(m) = f(m) - [f(m)] has the distribution function x. Theorem II (p. 2) treats sum_p f'(p)^2/p convergent and sum_p f'(p)/p divergent: f centred by the partial sums of f'(p)/p has a distribution function, continuous and strictly increasing on the whole line. As printed the centring sum runs over all p, which diverges, and the function said to have the distribution function is f(m); the statement is read with the sum truncated, as Theorem III (p. 2) writes it. Theorem II's proof is omitted. Theorem III gives a distribution function for f(m) - sum_(p<=n) f(p)/p + c when f(m) - c log m satisfies the hypotheses of Theorem II.

Theorems IV and V (pp. 2-3) are the structural results. Theorem V says that if for infinitely many n more than c_1 n integers up to n have f-values pairwise within c_2, then for some constant c, with f^+(p) = f(p) - c log p, the series sum_p (f^+)'(p)^2/p converges; the print writes the series as sum_p ((f^+)'(p)/p)^2, which converges for every c, while the converse stated on p. 3, the proof (pp. 8-14) and the later uses take the form given here. Such f are called finitely distributed. Theorem IV, deduced from it, says that when sum_(f(p) != 0) 1/p diverges, for every eps > 0 there is a delta > 0 such that fewer than eps n integers up to n have f-values in one interval of length delta, for n large; the print names the two constants the other way round, which as printed is trivial, and its proof (pp. 14-17) is for the order given here. Theorem VI (pp. 3-4) is a law of the iterated logarithm for the sums of f(p) over prime divisors p <= u of m, stated without proof. The proof of Theorem I uses the method of the Erdos-Kac paper with Berry's quantitative central limit theorem (Lemma 1, p. 6).

On p. 3 the paper says a result "probably holds" that it cannot prove: if f(n+1) - f(n) < c_1 for all n, then f(n) = c log n + phi(n) with |phi(n)| < c_2. It conjectures that f(n) = c log n when f(n+1) >= f(n) for all n outside a set of density 0, or when f(n+1) - f(n) -> 0 along a sequence of density 1. It proves the cases without exceptions: f(m) = c log m when f(m+1) >= f(m) for every m (Theorem XI, p. 17) and when f(m+1) - f(m) -> 0 (Theorem XIII, p. 18). Theorems VII to X, XII and XIV, and the complex-valued Theorems IV' and V' (p. 20), have no result pages here; most are stated without proof.

Source: https://users.renyi.hu/~p_erdos/1946-06.pdf.

Read status. Claims checked: Theorems I to VI, XI and XIII and the statements of p. 3 were read clause by clause on the page images of the print. The proofs of Theorems I, IV, V and XIII were read for structure and that of Theorem XI followed; Theorems II and VI and Theorem X, on which Theorem XI rests, are stated without proof in the paper. Nothing here is independently reviewed.

Bears on.

• #491: the statement on p. 3 (conjectures) is posed, not proved; its one-sided hypothesis f(n+1) - f(n) < c_1 is weaker than the problem's |f(n+1) - f(n)| < c, with the same conclusion. Theorem XI and Theorem XIII give the conclusion with error 0 when f is nondecreasing or f(n+1) - f(n) -> 0.
• #1122: the first conjecture on p. 3 is the problem's question, posed without proof; Theorem XI proves the case in which f never decreases.

Results.

• Theorem I (p. 1): f(p) -> 0 and sum_p f'(p)^2/p = infinity make the fractional part of f(m) uniformly distributed.
• Theorem II (p. 2): under sum_p f'(p)^2/p < infinity and sum_p f'(p)/p divergent, the centred f has a continuous strictly increasing distribution function.
• Theorem III (p. 2): the same after subtracting c log m.
• Theorem IV (p. 2): sum_(f(p) != 0) 1/p = infinity prevents a positive proportion of integers from having f-values in a short interval.
• Theorem V (p. 3): finitely distributed f satisfy sum_p (f^+)'(p)^2/p < infinity with f^+(p) = f(p) - c log p, and conversely.
• Theorem VI (pp. 3-4): a law of the iterated logarithm for bounded f.
• Theorem XI (p. 17): nondecreasing additive f are c log m.
• Theorem XIII (p. 18): f(m+1) - f(m) -> 0 gives f(m) = c log m.
• Conjectures (p. 3): the bounded-above-differences statement and the two density-zero-exception conjectures.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.