Weighted uniform consistency of kernel density estimators with general bandwidth sequences.

Dony, Julia; Einmahl, Uwe

Displaying similar documents to “Weighted uniform consistency of kernel density estimators with general bandwidth sequences.”

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Landau’s function for one million billions

Marc Deléglise, Jean-Louis Nicolas, Paul Zimmermann (2008)

Journal de Théorie des Nombres de Bordeaux

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Let $𝔖_{n}$ denote the symmetric group with $n$ letters, and $g (n)$ the maximal order of an element of $𝔖_{n}$ . If the standard factorization of $M$ into primes is $M = q_{1}^{α_{1}} q_{2}^{α_{2}} ... q_{k}^{α_{k}}$ , we define $ℓ (M)$ to be $q_{1}^{α_{1}} + q_{2}^{α_{2}} + ... + q_{k}^{α_{k}}$ ; one century ago, E. Landau proved that $g (n) = {max}_{ℓ (M) \leq n} M$ and that, when $n$ goes to infinity, $log g (n) \sim \sqrt{n log (n)}$ . There exists a basic algorithm to compute $g (n)$ for $1 \leq n \leq N$ ; its running time is $𝒪 (N^{3 / 2} / \sqrt{log N})$ and the needed memory is $𝒪 (N)$ ; it allows computing $g (n)$ up to, say, one million. We describe an algorithm to calculate $g (n)$ for $n$ up to $10^{15}$ . The main idea is to use the...

A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip $3 / 4 \leq σ < 1$

Yuk-Kam Lau (2006)

Journal de Théorie des Nombres de Bordeaux

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Let $E_{σ} (T)$ be the error term in the mean square formula of the Riemann zeta-function in the critical strip $1 / 2 < σ < 1$ . It is an analogue of the classical error term $E (T)$ . The research of $E (T)$ has a long history but the investigation of $E_{σ} (T)$ is quite new. In particular there is only a few information known about $E_{σ} (T)$ for $3 / 4 < σ < 1$ . As an exploration, we study its mean value $\int_{1}^{T} E_{σ} (u) d u$ . In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of $T$ .