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Structure and randomness in arithmetic settings Wong, Erick Bryce

Abstract

We study questions in three arithmetic settings, each of which carries aspects of random-like behaviour. In the setting of arithmetic functions, we establish mild conditions under which the tuple of multiplicative functions [f₁, f₂, …, f_d ], evaluated at d consecutive integers n+1, …, n+d, closely approximates points in R^d for a positive proportion of n; we obtain a further generalization which allows these functions to be composed with various arithmetic progressions. Secondly, we examine the eigenvalues of random integer matrices, showing that most matrices have no rational eigenvalues; we also identify the precise distributions of both real and rational eigenvalues in the 2 × 2 case. Finally, we consider the set S(k) of numbers represented by the quadratic form x² + ky², showing that it contains infinitely many strings of five consecutive integers under many choices of k; we also characterize exactly which numbers can appear as the difference of two consecutive values in S(k).

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Attribution-NonCommercial-NoDerivatives 4.0 International

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