Explicit estimates for the summatory function of Λ(n)/n from the one of Λ(n)

Olivier Ramaré

Displaying similar documents to “Explicit estimates for the summatory function of Λ(n)/n from the one of Λ(n)”

Estimates of $\sum_{k = 1}^{N} k^{- s} 〈 k x 〉^{- t}$

A. Kruse (1967)

Acta Arithmetica

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On the behaviour close to the unit circle of the power series with Möbius function coefficients

Oleg Petrushov (2014)

Acta Arithmetica

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Let $(z) = \sum_{n = 1}^{\infty} μ (n) z^{n}$ . We prove that for each root of unity $e (β) = e^{2 π i β}$ there is an a > 0 such that $(e (β) r) = Ω ({(1 - r)}^{- a})$ as r → 1-. For roots of unity e(l/q) with q ≤ 100 we prove that these omega-estimates are true with a = 1/2. From omega-estimates for (z) we obtain omega-estimates for some finite sums.

Uniform $L^{1}$ error bounds for semi-discrete finite element solutions of evolutionary integral equations

Lin, Qun, Xu, Da, Zhang, Shuhua

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In this paper, we consider the second-order continuous time Galerkin approximation of the solution to the initial problem $u_{t} + \int_{0}^{t} β (t - s) A u (s) d s = 0, u (0) = v, t > 0,$ where A is an elliptic partial-differential operator and $β (t)$ is positive, nonincreasing and log-convex on $(0, \infty)$ with $0 \leq β (\infty) < β (0^{+}) \leq \infty$ . Error estimates are derived in the norm of $L_{t}^{1} (0, \infty; L_{x}^{2})$ , and some estimates for the first order time derivatives of the errors are also given.

On certain functions that generalize von Mangoldt’s function $Λ (n)$

A. Ivić (1975)

Matematički Vesnik

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Gradient estimates for the p(x)-Laplacian equation in $ℝ^{N}$

Chao Zhang, Shulin Zhou, Bin Ge (2015)

Annales Polonici Mathematici

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Under some assumptions on the function p(x), we obtain global gradient estimates for weak solutions of the p(x)-Laplacian type equation in $ℝ^{N}$ .

The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type

Tie Zhu Zhang, Shu Hua Zhang (2015)

Applications of Mathematics

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We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of $C$ -uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set $S$ , the gradient approximation possesses the superconvergence: ${max}_{P \in S} | (\nabla u - \bar{\nabla} u_{h}) (P) | = O (h^{2}) {| ln h |}^{3 / 2}$ , where $\bar{\nabla}$ denotes the average gradient on elements containing vertex $P$ . Furthermore, by using the interpolation post-processing...

Lyapunov functions and $L^{p}$ -estimates for a class of reaction-diffusion systems

Dirk Horstmann (2001)

Colloquium Mathematicae

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We give a sufficient condition for the existence of a Lyapunov function for the system aₜ = ∇(k(a,c)∇a - h(a,c)∇c), x ∈ Ω, t > 0, $ε c ₜ = k_{c} Δ c - f (c) c + g (a, c)$ , x ∈ Ω, t > 0, for $Ω \subset ℝ^{N}$ , completed with either a = c = 0, or ∂a/∂n = ∂c/∂n = 0, or k(a,c) ∂a/∂n = h(a,c) ∂c/∂n, c = 0 on ∂Ω × t > 0. Furthermore we study the asymptotic behaviour of the solution and give some uniform $L^{p}$ -estimates.

Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $ℝ^{3}$

M. Burak Erdoğan, Michael Goldberg, Wilhelm Schlag (2008)

Journal of the European Mathematical Society

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We present a novel approach for bounding the resolvent of $H = - Δ + i (A \cdot \nabla + \nabla \cdot A) + V = : - Δ + L 1$ for large energies. It is shown here that there exist a large integer $m$ and a large number $λ_{0}$ so that relative to the usual weighted $L^{2}$ -norm, $∥ {(L {(- Δ + (λ + i 0))}^{- 1})}^{m} ∥ < \frac{1}{2} 2$ for all $λ > λ_{0}$ . This requires suitable decay and smoothness conditions on $A, V$ . The estimate (2) is trivial when $A = 0$ , but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...

Several notes on the circumradius condition

Václav Kučera (2016)

Applications of Mathematics

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Recently, the so-called circumradius condition (or estimate) was derived, which is a new estimate of the $W^{1, p}$ -error of linear Lagrange interpolation on triangles in terms of their circumradius. The published proofs of the estimate are rather technical and do not allow clear, simple insight into the results. In this paper, we give a simple direct proof of the $p = \infty$ case. This allows us to make several observations such as on the optimality of the circumradius estimate. Furthermore, we show how...

On some $L^{p}$ -estimates for solutions of elliptic equations in unbounded domains

Sara Monsurrò, Maria Transirico (2015)

Mathematica Bohemica

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In this review article we present an overview on some a priori estimates in $L^{p}$ , $p > 1$ , recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two $L^{p}$ -bounds, $p > 2$ , for...

Cutting the loss of derivatives for solvability under condition $(Ψ)$

Nicolas Lerner (2006)

Bulletin de la Société Mathématique de France

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For a principal type pseudodifferential operator, we prove that condition $(ψ)$ implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from $ϵ + 3 / 2$ for any $ϵ > 0$ (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition $(ψ)$ doesimply local solvability with a loss of 1...

Some characterizations of the class $_{m} (Ω)$ and applications

Hai Mau Le, Hong Xuan Nguyen, Hung Viet Vu (2015)

Annales Polonici Mathematici

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We give some characterizations of the class $_{m} (Ω)$ and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

An application of the method wherein the $p h i$ - and $c h i$ - methods are combined for the determination of the grating constant. [II.]

Swami Jnanananda (1936)

Časopis pro pěstování matematiky a fysiky

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The joint essential numerical range of operators: convexity and related results

Chi-Kwong Li, Yiu-Tung Poon (2009)

Studia Mathematica

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Let W(A) and $W_{e} (A)$ be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A₁, ..., Aₘ) acting on an infinite-dimensional Hilbert space. It is shown that $W_{e} (A)$ is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ 1, ..., m, $W_{e} (A)$ can be obtained as the intersection of all sets of the form $c l (W (A ₁, . . ., A_{i + 1}, A_{i} + F, A_{i + 1}, . . ., A ₘ))$ , where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in...

Nonanalyticity of solutions to $\partial_{t} u = \partial ²_{x} u + u ²$

Grzegorz Łysik (2003)

Colloquium Mathematicae

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It is proved that the solution to the initial value problem $\partial_{t} u = \partial ²_{x} u + u ²$ , u(0,x) = 1/(1+x²), does not belong to the Gevrey class $G^{s}$ in time for 0 ≤ s < 1. The proof is based on an estimation of a double sum of products of binomial coefficients.