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The least common multiple of a quadratic sequence. (English) Zbl 1248.11068

Let \(f\) be an irreducible quadratic polynomial. The author studies the least common multiple of the integers \(f(1), f(2), \ldots, f(n)\), and proves the following: We have \[ \log \mathrm{l.c.m} (f(1), \ldots, f(n) = n\log n + Bn+o(n), \] where \(B\) is a constant depending on \(f\), for which a quite complicated expression is derived. The author first indicates that the same result with \(O(n)\) in place of \(o(n)\) can be obtained in a straightforward manner, the error term essentially coming from rounding errors. To obtain \(o(n)\) one has to obtain some cancellation between these rounding errors. To do so the author shows that as \(p\) varies, the roots of \(f\) modulo \(p\) are equidistributed. Proving this equidistribution forms the bulk of the paper.

MSC:

11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11B83 Special sequences and polynomials
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References:

[1] doi:10.2307/2118527 · Zbl 0840.11003
[4] doi:10.1155/S1073792800000404 · Zbl 1134.11339
[5] doi:10.1007/BF03027290 · Zbl 0885.11005
[6] doi:10.2307/2695513 · Zbl 1124.11300
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