Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function

Let Δ(x) and E(t) denote respectively the remainder terms in the Dirichlet divisor problem and the mean square formula for the Riemann zeta-function on the critical line. This article is a survey of recent developments on the research of these famous error terms in number theory. These include upper bounds, Ω-results, sign changes, moments and distribution, etc. A few open problems are also discussed. © 2010 Science China Press and Springer-Verlag Berlin Heidelberg.postprin

On Roth's theorem concerning a cube and three cubes of primes

In this paper, we prove that with at most O (N1271/1296+ε) exceptions, all positive integers up to N are the sum of a cube and three cubes of primes. This improves an earlier result O (N169/170) of the first author and the classical result O(NL-A) of Roth.preprin

Large values of error terms of a class of arithmetical functions

We consider the error terms of a class of arithmetical functions whose Dirichlet series satisfy a functional equation with multiple gamma factors. Our aim is to establish Ω± results to a subclass of these arithmetical functions with a good localization of the occurrence of the extreme values. As applications, we improve the Ω± results of some special 3-dimensional ellipsoids of other writers and extend our result to other ellipsoids.published_or_final_versio

On the mean square formula of the error term in the Dirichlet divisor problem

Let F(x) be the remainder term in the mean square formula of the error term (t) in the Dirichlet divisor problem. We improve on the upper estimate of F(x) obtained by Preissmann around twenty years ago. The method is robust, which applies to the same problem for the error terms in the circle problem and the mean square formula of the Riemann zeta-function. © Cambridge Philosophical Society 2008.postprin

Nutritional status of young children with inherited blood disorders in western Kenya.

To determine the association between a range of inherited blood disorders and indicators of poor nutrition, we analyzed data from a population-based, cross-sectional survey of 882 children 6–35 months of age in western Kenya. Of children with valid measurements, 71.7% were anemic (hemoglobin < 11 g/dL), 19.1% had ferritin levels < 12 μg/L, and 30.9% had retinol binding protein (RBP) levels < 0.7 μmol/L. Unadjusted analyses showed that compared with normal children, homozygous α(+)-thalassemia individuals had a higher prevalence of anemia (82.3% versus 66.8%, P = 0.001), but a lower prevalence of low RBP (20.5% versus 31.4%, P = 0.024). In multivariable analysis, homozygous α(+)-thalassemia remained associated with anemia (adjusted odds ratio [aOR] = 1.8, P = 0.004) but not with low RBP (aOR = 0.6, P = 0.065). Among young Kenyan children, α(+)-thalassemia is associated with anemia, whereas G6PD deficiency, haptoglobin 2-2, and HbS are not; none of these blood disorders are associated with iron deficiency, vitamin A deficiency, or poor growth

Conditional bounds for small prime solutions of linear equations

Let a 1, a 2, a 3 be non-zero integers with gcd(a 1 a 2, a 3)=1 and let b be an arbitrary integer satisfying gcd (b, a i, a j) =1 for i≠j and b≡a 1+a 2+a 3 (mod 2). In a previous paper [3] which completely settled a problem of A. Baker, the 2nd and 3rd authors proved that if a 1, a 2, a 3 are not all of the same sign, then the equation a 1 p 1+a 2 p 2+a 3 p 3=b has a solution in primes p j satisfying {Mathematical expression} where A>0 is an absolute constant. In this paper, under the Generalized Riemann Hypothesis, the authors obtain a more precise bound for the solutions p j . In particular they obtain A0. An immediate consquence of the main result is that the Linnik's courtant is less than or equal to 2. © 1992 Springer-Verlag.postprin

An extension to the Brun-Titchmarsh theorem

The Siegel-Walfisz theorem states that for any B > 0, we have ∑/p≤x/p≡a(mod k) 1 ∼ x/φ(k) lox x for k ≤ log B x and (k, a) = 1. This only gives an asymptotic formula for the number of primes over an arithmetic progression for quite small moduli k compared with x. However, if we are only concerned about upper bound, we have the Brun-Titchmarsh theorem, namely for any 1 ≤ k 0, s ≥ 1 and 1 ≤ k < x.In particular, for s ≤ log log (x/k), we have ∑/y<n≤x+y ≡ a (mod k)ω (n) < s 1 ≪ x/φ (k) log (x/k) (log log (x/k) + K)s-1/(s-1)! √ log log (x/k) + K and for any ε∈(0, 1) and s ≤ (1-ε) log log (x/k), we have. ∑/y<n≤x+y ≡ a (mod k)ω (n) < s 1 ≪ ε-1x/φ (k) log (x/k) (log log (x/k) +K)s-1/(s-1) !. © 2010. Published by Oxford University Press. All rights reserved.postprin

Optical studies of ZnS:Mn films grown by pulsed laser deposition

Author name used in this publication: C. L. MakAuthor name used in this publication: K. H. Wong2002-2003 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Raf/MEK/MAPK signaling stimulates the nuclear translocation and transactivating activity of FOXM1c

The forkhead box (FOX) transcription factor FOXM1 is ubiquitously expressed in proliferating cells. FOXM1 expression peaks at the G2/M phase of the cell cycle and its functional deficiency in mice leads to defects in mitosis. To investigate the role of FOXM1 in the cell cycle, we used synchronized hTERT-BJ1 fibroblasts to examine the cell cycle-dependent regulation of FOXM1 function. We observed that FOXM1 is localized mainly in the cytoplasm in cells at late-G1 and S phases. Nuclear translocation occurs just before entry into the G2/M phase and is associated with phosphorylation of FOXM1. Consistent with the dependency of FOXM1 function on mitogenic signals, nuclear translocation of FOXM1 requires activity of the Raf/MEK/MAPK signaling pathway and is enhanced by the MAPK activator aurintricarboxylic acid. This activating effect was suppressed by the MEK1/2 inhibitor U0126. In transient reporter assays, constitutively active MEK1 enhances the transactivating effect of FOXM1c, but not FOXM1b, on the cyclin B1 promoter. RT-PCR analysis confirmed that different cell lines and tissues predominantly express the FOXM1c transcript. Mutations of two ERK1/2 target sequences within FOXM1c completely abolish the MEK1 enhancing effect, suggesting a direct link between Raf/MEK/MAPK signaling and FOXM1 function. Importantly, inhibition of Raf/MEK/MAPK signaling by U0126 led to suppression of FOXM1 target gene expression and delayed progression through G2/M, verifying the functional relevance of FOXM1 activation by MEK1. In summary, we provide the first evidence that Raf/MEK/MAPK signaling exerts its G2/M regulatory effect via FOXM1c.published_or_final_versio

The effects of stent porosity on the endovascular treatment of intracranial aneurysms located near a bifurcation

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