Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 50

Statement. Schoenberg proved that for every $c\in [0,1]$ the density of

\{ n\in \mathbb{N} : \phi(n)<cn\}

exists. Let this density be denoted by $f(c)$ . Is it true that there are no $x$ such that $f'(x)$ exists and is positive?

Status. Open.

Source. erdosproblems.com/50, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #50, https://www.erdosproblems.com/50.

References.

[Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (6)

Arithmetic Functions Arithmetic Functions arithmetic_functions/erdos_1974_distribution_numbers_form_sigma_n_n Question (pp. 59-60): can the singular distribution function of sigma(n)/n have a derivative other than 0?Theorem (p. 60): fewer than c_1 x / log t integers up to x have sigma(n)/n in [a, a + 1/t)number_theory/erdos_1995_my_favourite_problems_number_theory_combinatorics

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