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The distribution of Fourier coefficients of cusp forms over sparse sequences

Huixue Lao, Ayyadurai Sankaranarayanan (2014)

Acta Arithmetica

Let λ f ( n ) be the nth normalized Fourier coefficient of a holomorphic Hecke eigenform f ( z ) ∈ S k ( Γ ) . We establish that ∑ n ≤ x λ f 2 ( n j ) = c j x + O ( x 1 - 2 / ( ( j + 1 ) 2 + 1 ) ) for j = 2,3,4, which improves the previous results. For j = 2, we even establish a better result.

The distribution of powers of integers in algebraic number fields

Werner Georg Nowak, Johannes Schoißengeier (2004)

Journal de Théorie des Nombres de Bordeaux

For an arbitrary (not totally real) number field K of degree ≥ 3 , we ask how many perfect powers γ p of algebraic integers γ in K exist, such that μ ( τ ( γ p ) ) ≤ X for each embedding τ of K into the complex field. ( X a large real parameter, p ≥ 2 a fixed integer, and μ ( z ) = max ( | Re ( z ) | , | Im ( z ) | ) for any complex z .) This quantity is evaluated asymptotically in the form c p , K X n / p + R p , K ( X ) , with sharp estimates for the remainder R p , K ( X ) . The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result on the approximation...

The distribution of second p -class groups on coclass graphs

Daniel C. Mayer (2013)

Journal de Théorie des Nombres de Bordeaux

General concepts and strategies are developed for identifying the isomorphism type of the second p -class group G = Gal ( F p 2 ( K ) | K ) , that is the Galois group of the second Hilbert p -class field F p 2 ( K ) , of a number field K , for a prime p . The isomorphism type determines the position of G on one of the coclass graphs 𝒢 ( p , r ) , r ≥ 0 , in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field K and of its p -class group Cl p ( K ) , the position of G is restricted to certain admissible branches of coclass...

Currently displaying 301 – 320 of 1340