Oscillatory properties of $M(x) = \sum _{n≤x} μ(n)$, I

J. Pintz

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The half dimensional sieve

Henryk Iwaniec (1976)

Acta Arithmetica

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Oscillatory properties of $M (x) = \sum_{n \leq x} μ (n)$ , III

J. Pintz (1984)

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On sums of Hecke series in short intervals

Aleksandar Ivić (2001)

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We have $\sum_{K - G \leq k_{j} \leq K + G} α_{j} H_{j}^{3} (\frac{1}{2}) ≪_{ϵ} G K^{1 + ϵ}$ for $K^{ϵ} \leq G \leq K, where α_{j} = {|ρ_{j} (1)|}^{2} {(cosh π k_{j})}^{- 1}, and ρ_{j} (1)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue $λ_{j} = k_{j}^{2} + \frac{1}{4}$ to which the Hecke series $H_{j} (s)$ is attached. This result yields the new bound $H_{j} (\frac{1}{2} ≪_{ϵ} k_{j}^{\frac{1}{3} + ϵ} .$