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Matrix coefficients, counting and primes for orbits of geometrically finite groups

Amir Mohammadi, Hee Oh (2015)

Journal of the European Mathematical Society

Let G : = SO ( n , 1 ) ∘ and Γ ( n - 1 ) / 2 for n = 2 , 3 and when δ > n - 2 for n ≥ 4 , we obtain an effective archimedean counting result for a discrete orbit of Γ in a homogeneous space H ∖ G where H is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family { ℬ T ⊂ H ∖ G } of compact subsets, there exists η > 0 such that # [ e ] Γ ∩ ℬ T = ℳ ( ℬ T ) + O ( ℳ ( ℬ T ) 1 - η ) for an explicit measure ℳ on H ∖ G which depends on Γ . We also apply the affine sieve and describe the distribution of almost primes on orbits of Γ in arithmetic settings....

Mean values connected with the Dedekind zeta-function of a non-normal cubic field

Guangshi Lü (2013)

Open Mathematics

After Landau’s famous work, many authors contributed to some mean values connected with the Dedekind zetafunction. In this paper, we are interested in the integral power sums of the coefficients of the Dedekind zeta function of a non-normal cubic extension K 3/ℚ, i.e. S l , K 3 ( x ) = ∑ m ⩽ x M l ( m ) , where M(m) denotes the number of integral ideals of the field K 3 of norm m and l ∈ ℕ. We improve the previous results for S 2 , K 3 ( x ) and S 3 , K 3 ( x ) .

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