Erdős Problem #950

The question

erdos_950.parts.i. Is it true that $\mathrm{liminf}f(n)=1$?

erdos_950.parts.ii. Is it true that $\mathrm{limsup}f(n)=\infty$?

erdos_950.parts.iii. Is it true that $f(n)=o(\log \log n)$ for all $n$?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«950» (3 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_950.parts.i :
    answer(sorry) ↔ atTop.liminf (fun n : ℕ ↦ (f n : EReal)) = 1

theorem erdos_950.parts.ii :
    answer(sorry) ↔ atTop.limsup (fun n : ℕ ↦ (f n : EReal)) = ⊤

theorem erdos_950.parts.iii :
    answer(sorry) ↔ f =o[atTop] (fun n : ℕ ↦ Real.log (Real.log n))

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already — check erdosproblems.com/950. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

• erdos_950.variants.weaker_pi — Erdős writes that a 'weaker conjecture which is perhaps not quite inaccessible' is that, for every ε>0, if x is sufficiently large there exists y<x such that…
• erdos_950.variants.sum_primes — The study of f(p) is even harder, and Erdős could not prove that Σ_p<xf(p)^2∼ π(x).

References

• erdosproblems.com/855
• erdosproblems.com/950
• [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
• mathoverflow/508491

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.