In their 1984 paper Asymptotic of the Largest and the Typical Dimensions of Irreducible Representations of a Symmetric Group, Vershik and Kerov use the notation $\DeclareMathOperator{\arch}{arch}\arch x$ to denote some function. This function is crucial in the proof, but I have not been able to figure out what it is. I think it first appears in formula (8) on page 25. I will really appreciate any hints you have to give! Thank you!
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