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On the Lower Range of Perron's Modular Function
Published online by Cambridge University Press: 20 November 2018
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The modular function Mwas introduced by Perron in (6). M(ξ) (for irrational ξ) is denned by the property that the inequality
is satisfied by an infinity of relatively prime pairs (p, q)for positive d,but by at most a finite number of such pairs for negative d.We will write
for the continued fraction expansion of ξ ∈ (0, 1) and for any finite collection y1,…, ykof positive integers we will write
It is known (see 6) that
Where
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