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 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

On a Waring - Goldbach-type problem for fourth powers

In this paper, we prove that every sufficiently large positive integer satisfying some necessary congruence conditions can be represented by the sum of a fourth power of integer and twelve fourth powers of prime numbers. © 2004 Elsevier Inc. All rights reserved.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

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

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

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

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