# What is 'arch' in Vershik-Kerov's 1984 paper?

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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A search in MathSciNet showed me a unique paper with that title and those authors, but it was dated 1981, not 1984. An English translation is available (thanks to my university's Springer subscription), but it is only 12 pages long and its formula (8) contains no arch. Perhaps the paper you have is a later, expanded version that didn't get into Math Reviews. –  Andreas Blass Dec 21 '11 at 18:36
Sorry, you are right. The title is Asymptotic of the Largest and the Typical Dimensions of Irreducible Representations of a Symmetric Group. I will correct it above... –  Zatrapilla Dec 21 '11 at 18:49
I believe it is the inverse hyperbolic cosine, although it's been years since I read this paper so I may be off about this. –  Henry Cohn Dec 21 '11 at 19:21
Henry Cohn's guess is correct. –  Fedor Petrov Dec 21 '11 at 19:45
On the page ru.wikipedia.org/wiki/Гиперболические_функции, down near the bottom after the second graph plot there is a listing of inverse hyperbolic functions in Russian notation and Arch x is the second on the list. –  KConrad Dec 21 '11 at 22:47
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