A lower bound for the error term in Weyl’s law for certain Heisenberg manifolds, II

Werner Nowak

Displaying similar documents to “A lower bound for the error term in Weyl’s law for certain Heisenberg manifolds, II”

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Manfred Kühleitner, Werner Nowak (2006)

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The arithmetic function ρ(n) counts the number of ways to write a positive integer n as a difference of two squares. Its average size is described by the Dirichlet summatory function Σn≤x ρ(n), and in particular by the error term R(x) in the corresponding asymptotics. This article provides a sharp lower bound as well as two mean-square results for R(x), which illustrates the close connection between ρ(n) and the number-of-divisors function d(n).

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Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of $|ς (\frac{1}{2} + i t)|$ . If $E^{*} (t) = E (t) - 2 π Δ^{*} (t / 2 π)$ with $Δ^{*} (x) = - Δ (x) + 2 Δ (2 x) - \frac{1}{2} Δ (4 x)$ , then we obtain $\int_{0}^{T} {(E^{*} (t))}^{4} d t ≪_{e} T^{16 / 9 + ε}$ . We also show how our method of proof yields the bound $\sum_{r = 1}^{R} {(\int_{t r - G}^{t r + G} {|ς (\frac{1}{2} + i t)|}^{2} d t)}^{4} ≪_{e} T^{2 + e} G^{- 2} + R G^{4} T^{ε}$ , where T 1/5+ε≤G≪T, T

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On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

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The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.

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On a generalization of a formula of Sierpiński.

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