Mertens function M(n) = Σμ(k) = μ(1)+μ(2)+...+μ(n). M(1)=1, M(2)=0, M(3)=−1, M(4)=−1, M(5)=−2... The Mertens conjecture (|M(n)|<√n for all n>1) was disproved in 1985, but the first counterexample is > 10^{10^{13}}!
How to Use
Enter n:
- M(n): Cumulative sum
- μ(k): Individual values
- √n bound: Mertens comparison
The Disproof
Mertens conjectured |M(n)|<√n (1897). Odlyzko & te Riele (1985) disproved it — but the first counterexample hasn't been found explicitly! Only its existence is proven. The actual violation occurs at an astronomically large n.
Riemann Connection
RH ⟺ M(n) = O(n^{1/2+ε}) for all ε>0. So M(n) grows at most like n^{0.5+tiny}. If M(n) could be bounded by n^{0.5−δ} for some δ>0, that would prove more than RH!
Step-by-Step Instructions
- 1Enter n.
- 2Compute M(n).
- 3See μ values.
- 4Compare √n.
- 5Check growth.