/-
Copyright 2025 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
import FormalConjecturesUtilErdős Problem 259
open scoped ArithmeticFunction.Moebius
namespace Erdos259
Is $\sum_{n} \mu(n)^2\frac{n}{2^n}$ irrational?
This is true, and was proved by Chen and Ruzsa.
[ChRu99] Chen, Yong-Gao and Ruzsa, Imre Z., On the irrationality of certain series. Period. Math. Hungar. (1999), 31--37.
@[category research solved, AMS 11, formal_proof using lean4 at "https://gist.githubusercontent.com/ster-oc/c7429943f6b3a634797dc8b2a3b01f2d/raw/8c6b5b7f08021f0aed2312542dd2e9ee7beaa6d6/Erdos259.lean"]
theorem erdos_259 : Irrational (∑' n : ℕ, (μ n) ^ 2 * n / (2 ^ n)) := ⊢ Irrational (∑' (n : ℕ), ↑(μ n) ^ 2 * ↑n / 2 ^ n)
All goals completed! 🐙
end Erdos259