Erdős problem 897 · problems.science

Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$ such that $\limsup_{p,k} f(p^k) / \log(p^k) = ∞$ . Is it true that $\limsup_n (f(n+1)−f(n))/ \log n = ∞$ ?

Sources

Browse retained paths and inspect the exact material available for this Problem.

5 retained statements2415f78e850a

Open selected source

FormalConjectures/ErdosProblems/

897.lean

Retained formal statement3 of 5

Wirsing [Wi70] proved that if $|f(n+1)−f(n)| ≤ C$ then $f(n) = c \log n + O(1)$ for some constant $c$ .

FormalConjectures/ErdosProblems/897.leanErdos897.erdos_897.variants.log_growth3 linesExact file

1∀ (f : ℕ → ℝ),2  (∀ a > 0, ∀ b > 0, a.Coprime b → f (a * b) = f a + f b) →3    ∀ (C : ℝ), (∀ (n : ℕ), |f (n + 1) - f n| ≤ C) → ∃ c O, O =O[Filter.atTop] 1 ∧ ∀ (n : ℕ), f n ≤ c * Real.log ↑n + O n

SolvedStatement only, no proof

Sources

Search problems.science