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We show that the length of the set  \[    { z ∈  C : | ∏  i = 1 n ( z −  α  i ) | = 1 } \left \{ {z \in \mathbb {C}:|\prod \limits _{i = 1}^n {(z - {\alpha _i})| = 1} } \right \}     \]  is at most      8 π  e n 8\pi en      . This gives the correct rate of growth in a long-standing open problem of Erdös, Herzog, and Piranian and improves the previous bound of      74 n 2 74{n^2}       due to Pommerenke.

The arc length of the lemniscate $\{\vert p(z)\vert =1\}$