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. $$ S_{l,K_3 } (x) = \sum\nolimits_{m \leqslant x} {M^l (m)} $$, 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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Let $λ_f(n)$ be the nth normalized Fourier coefficient of a holomorphic or Maass cusp form f for SL(2,ℤ). We establish the asymptotic formula for the summatory function $∑_{\substack{n≤x\\ n≡l (mod q)}} |λ_f(n)|^{2j}$ as x → ∞, where q grows with x in a definite way and j = 2,3,4.
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