FormalConjectures.ErdosProblems.«15»

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module

public import FormalConjecturesUtil

Erdős Problem 15: Convergence of Series with Primes

Reference: erdosproblems.com/15

@[expose] public sectionnamespace Erdos15open Filter Topology

Is it true that $\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}$ converges, where $p_n$ is the sequence of primes?

Note: In the problem statement, $p_n$ is the $n$-th prime, indexed such that $p_1=2, p_2=3, \ldots$. We 0-index here to reflect how Nat.nth works.

Note: convergence here is convergence of the sequence of partial sums, which is what the problem asks about. Summable would be the wrong notion: it is unconditional summability, equivalent over $\mathbb{R}$ to absolute convergence, and $\sum_n n/p_n$ diverges.

@[category research open, AMS 11]
theorem erdos_15 : answer(sorry) ↔
    ∃ l : ℝ, Tendsto (fun N => ∑ k ∈ Finset.range N,
      (-1 : ℝ) ^ (k + 1) * (k + 1) / (k.nth Nat.Prime)) atTop (𝓝 l) := by⊢ True ↔ ∃ l, Tendsto (fun N ↦ ∑ k ∈ Finset.range N, (-1) ^ (k + 1) * (↑k + 1) / ↑(Nat.nth Nat.Prime k)) atTop (𝓝 l)
  sorryAll goals completed! 🐙

-- TODO: add the other statements from the additional material


end Erdos15