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Mahler's 3/2 problem

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In mathematics, Mahler's 3/2 problem concerns the existence of "Z-numbers".

A Z-number is a real number x such that the fractional parts of

are less than 1/2 for all positive integers n. Kurt Mahler conjectured in 1968 that there are no Z-numbers.

More generally, for a real number α, define Ω(α) as

Mahler's conjecture would thus imply that Ω(3/2) exceeds 1/2. Flatto, Lagarias, and Pollington showed[1] that

for rational p/q > 1 in lowest terms.

References

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  1. ^ Flatto, Leopold; Lagarias, Jeffrey C.; Pollington, Andrew D. (1995). "On the range of fractional parts of ζ { (p/q)n }". Acta Arithmetica. LXX (2): 125–147. doi:10.4064/aa-70-2-125-147. ISSN 0065-1036. Zbl 0821.11038.