* Preliminaries* Numbers* Sequences and series* Limits and continuity* Differentiation* The elementary transcendental functions* Integration* Infinite series and infinite products* Trigonometric series* Bibliography* Other works by the author* Index

An introduction to classical real analysis, by Karl R. Stromberg. Pp 575. $29·95. 1981. ISBN 0-534-98012-0 (Wadsworth)

The paper contains several characterizations of Banach lattices $E$ with the dual positive Schur property (i.e., $0 \le f_n \xrightarrow {\sigma (E^*,E)} 0$ implies $\Vert f_n\Vert \rightarrow 0$) and various examples of spaces having this property. We also investigate relationships between the dual positive Schur property, the positive Schur property, the positive Grothendieck property and the weak Dunford–Pettis property.

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On the dual positive Schur property in Banach lattices

An infinite family ${\fancyscript{A}}\subset {\fancyscript{P}}(\omega )$ is almost disjoint (AD) if the intersection of any two distinct elements of ${\fancyscript{A}}$ is finite. We survey recent results on almost disjoint families, concentrating on their applications to topology.

Almost Disjoint Families and Topology

We prove a general results on complemented unconditional basic sequences in Banach lattices and apply it to give some new examples of spaces with unique unconditional basis We show that Tsirelson space and certain Nakano spaces have unique unconditional bases We also construct an example of a space with a unique unconditional basis with a complemented subspace failing to have a unique unconditional basis

Uniqueness of unconditional bases in Banach spaces

We prove a general result on complemented unconditional basic sequences in Banach lattices and apply it to give some new examples of spaces with unique unconditional basis. We show that Tsirelson space and certain Nakano spaces have the unique unconditional bases. We also construct an example of a space with a unique unconditional basis with a complemented subspace failing to have a unique unconditional basis.

From the characterisation of geometrically convex and geometrically concave functions defined on (0,A] or $ [A,\infty) $ with $ A > 0 $, by means of their multiplicative conditions, we obtain unified proofs of some known and new inequalities. Functions of class C2 and strictly increasing on (a,b) fulfil some kind of supermultiplicativity and superadditivity. We have obtained a new constant determining the intervals of sub- and supermultiplicativity for the log function.

Multiplicative properties of real functions with applications to classical functions

Let $f(n)=\sigma(n)/e^\gamma n\log\log n$, $n=3,4,\ldots$ , where $\sigma$ denotes the sum of divisors function. In 1984 Robin proved that the inequality $f(n)>1$, for all $n\ge 5041$, is equivalent to the Riemann hypothesis. Here we show that the values of $f$ are close to $0$ on a set of asymptotic density $1$. Similarly, an inequality by Rosser and Schoenfeld of 1962, dealing with the Euler totient function $\varphi$, is essential only on "thin" subsets of $\mathbf{N}$.

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Robin's inequality and the Riemann hypothesis

Let M be a positive integer with M> 4, and let ϕ denote Euler’s totientfunction. If a positive integer n satisﬁes the Diophantine equation ( ∗ ) Mϕ ( n )= n− 1, then thenumber of prime factors of n is much bigger than M . Moreover, the set of all squarefree integerswhich do not fulﬁl ( ∗ ) contains “nice” subsets.Key words: Lehmer problem; Euler totient function.Let ϕ denote Euler’s totient function, and let M,n denote positive integers with M ≥ 2. Thepresent paper concerns the equation( ∗ ) Mϕ ( n )= n− 1considered in 1932 by Lehmer [5]. Throughout thispaper, for ﬁxed M the symbol L M stands for the(possibly empty) set of all composite solutionsof ( ∗ ),and we put L =˝ M≥ 2 L M . The letter n will alwaysdenote an element of L .By Q = {Q 1 ,Q 2 ,...} wedenote the set of all primes with Q i <Q i +1 ,where i =1 , 2 ,... .Lehmer asked whether the set L is nonempty,and he proved that if L=∅ then every n ∈L mustbe odd, squarefree, i.e.,(1) n = p 1 p 2 ·...·p r , with 3 ≤p 1

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On a Lehmer problem concerning Euler's totient function

We give a simple criterion for the solvability in integers of the negative Pell equation $x^2 - dy^2 = -1$ by means of primitive Pythagorean triples The method of proof allows us to also solve the problem of determinig the fundamental units of negative norm in real quadratic fields

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The negative Pell equation and Pythagorean triples

Let X, Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R: X → Y such that its range, R(X), is dense in Y.

Marek Wójtowicz

Papers

Multiplicative properties of real functions with applications to classical functions

Robin's inequality and the Riemann hypothesis

On a Lehmer problem concerning Euler's totient function

The negative Pell equation and Pythagorean triples

The controlled separable projection property for Banach spaces