Displaying similar documents to “Structural Theorems for Quasiasymptotics of Distributions at Infinity”

Sign changes of error terms related to arithmetical functions

Paulo J. Almeida (2007)

Journal de Théorie des Nombres de Bordeaux

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Let H ( x ) = ∑ n ≤ x φ ( n ) n - 6 π 2 x . Motivated by a conjecture of Erdös, Lau developed a new method and proved that # { n ≤ T : H ( n ) H ( n + 1 ) < 0 } ≫ T . We consider arithmetical functions f ( n ) = ∑ d ∣ n b d d whose summation can be expressed as ∑ n ≤ x f ( n ) = α x + P ( log ( x ) ) + E ( x ) , where P ( x ) is a polynomial, E ( x ) = - ∑ n ≤ y ( x ) b n n ψ x n + o ( 1 ) and ψ ( x ) = x - ⌊ x ⌋ - 1 / 2 . We generalize Lau’s method and prove results about the number of sign changes for these error terms.

Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Matthew H. Baker, Robert Rumely (2006)

Annales de l’institut Fourier

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Given a rational function ϕ ( T ) on ℙ 1 of degree at least 2 with coefficients in a number field k , we show that for each place v of k , there is a unique probability measure μ ϕ , v on the Berkovich space ℙ Berk , v 1 / ℂ v such that if { z n } is a sequence of points in ℙ 1 ( k ¯ ) whose ϕ -canonical heights tend to zero, then the z n ’s and their Gal ( k ¯ / k ) -conjugates are equidistributed with respect to μ ϕ , v . The proof uses a polynomial lift F ( x , y ) = ( F 1 ( x , y ) , F 2 ( x , y ) ) of ϕ to construct a two-variable Arakelov-Green’s function g ϕ , v ( x , y ) for each v . The measure μ ϕ , v is...