American Mathematical Monthly - February 2018

The February issue of the Monthly is packed full of interesting mathematics. The issue leads off with “The Circle Problem of Gauss and the Divisor Problem of Dirichlet—Still Unsolved,” where Bruce Berndt, Sun Kim, and Alexandru Zaharescu survey what is known about the errors involved in the asymptotic formulas for two famous arithmetic functions. Michael Maltenfort studies “Pascal functions,” that is, functions f defined on subsets of the 2-dimensional integer lattice that satisfy the rule f(a+1, b+1) = f(a, b) + f(a, b+1) that holds for Pascal’s triangle, recovers some old combinatorial identities, and discovers new ones. Bryan Dawson uses the hyperreals to construct an extension of the Riemann integral for which every bounded function is integrable, among other properties. Marcin Kulczycki, Dominik Kwietniak, and Jian Li define the notion of entropy of a shift space (i.e., a set of infinite sequences of symbols taken from a finite alphabet together with a set of prohibited finite blocks) and give an elementary proof that there is a shift space having an arbitrary nonnegative real number value as its entropy. Cara Brooks and Alberto Condori give a criterion (in terms of the norm of the resolvent) for when a square matrix A is normal.

In the Notes, Aaron Melman considers the Cauchy radius, i.e., the largest magnitude of a zero of a complex polynomial, how that radius can be reduced by means of a polynomial multiplier, and a class of polynomials for which that multiplier is optimal. Rom Pinchasi looks at a family consisting of an odd number of open unit-length intervals of R and shows that there will always be points that belong to an odd number of intervals in the family. Ralph Howard, Virginia Johnson, and George McNulty show that an ordered field is Archimedean precisely when every continuous additive function from the field to itself is linear. Christopher Stuart gives an appealing inequality for |sin(n)|, where n is a positive integer. Finally, Wenchang Chu gives a new proof of the Riesz interpolation formula using partial fraction decomposition.

Solve some problems with your special Valentine. And consider reading the late Raymond Smullyan’s A Beginner’s Guide to Mathematical Logic and his Beginner’s Further Guide to Mathematical Logic, both of which are reviewed by Robert Cowen.

— Susan Jane Colley, Editor

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Table of Contents

The Circle Problem of Gauss and the Divisor Problem of Dirichlet—Still Unsolved

p. 99.

Bruce C. Berndt, Sun Kim & Alexandru Zaharescu

Let r₂(n) denote the number of representations of the positive integer n as a sum of two squares, and let d(n) denote the number of positive divisors of n. Gauss and Dirichlet were evidently the first mathematicians to derive asymptotic formulas for ∑_n ⩽ _xr₂(n) and ∑_n ⩽ _xd(n), respectively, as x tends to infinity. But what is the error made in such approximations? Number theorists have been attempting to answer these two questions for over one and one-half centuries, and although we think that we essentially “know” what these errors are, progress in proving these conjectures has been agonizingly slow. Ramanujan had a keen interest in these problems, and although, to the best of our knowledge, he did not establish any bounds for the error terms, he did give us identities that have been used to derive bounds, and two further identities that might be useful, if we can figure out how to use them. In this paper, we survey what is known about these two famous unsolved problems, with a moderate emphasis on Ramanujan's contributions.

DOI: 10.1080/00029890.2018.1401853

Pascal Functions

p. 115.

Michael Maltenfort

We define Pascal functions by adapting the arithmetic rule that creates the Pascal triangle. By developing and applying properties of Pascal functions, we discover new identities and find new perspectives of old identities. The identities all involve binomial coefficients, with some also involving Stirling numbers, Stirling polynomials, associated Stirling numbers of the second kind, or Bell numbers.

DOI: 10.1080/00029890.2018.1401831

A New Extension of the Riemann Integral

p. 130.

C. Bryan Dawson

Using an equivalence relation on the hyperreals called approximation, a new extension of the Riemann integral is motivated and introduced in which every bounded function is integrable and for which there exists a function g: [0, 1] → R simultaneously satisfying (1) g is integrable, (2) g is unbounded on every subinterval of [0, 1], and (3) g is identical to its average value function.

DOI: 10.1080/00029890.2018.1401832

Entropy of Subordinate Shift Spaces

p. 141.

Marcin Kulczycki, Dominik Kwietniak & Jian Li

We introduce a new family of shift spaces—the subordinate shifts. Using subordinate shifts, we prove in an elementary way that for every nonnegative real number t there is a shift space with entropy t.

DOI: 10.1080/00029890.2018.1401875

A Resolvent Criterion for Normality

p. 149.

Cara D. Brooks & Alberto A. Condori

Given a normal matrix A and an arbitrary square matrix B (not necessarily of the same size), what relationships between A and B, if any, guarantee that B is also a normal matrix? We provide an answer to this question in terms of pseudospectra and norm behavior. In doing so, we prove that a certain distance formula, known to be a necessary condition for normality, is in fact sufficient and demonstrates that the spectrum of a matrix can be used to recover the spectral norm of its resolvent precisely when the matrix is normal. These results lead to new normality criteria and other interesting consequences.

DOI: 10.1080/00029890.2018.1401855

NOTES

Optimality of a Polynomial Multiplier

p. 158.

Aaron Melman

The Cauchy radius of a polynomial is a classical upper bound from 1829 on the magnitude of the largest zero of a polynomial, which was improved in 2002 by Rahman and Schmeisser with the use of a polynomial multiplier. Although this multiplier is not optimal for a general polynomial, we identify a large class of polynomials for which it is.

DOI: 10.1080/00029890.2018.1401878

A Theorem on Unit Segments on the Real Line

p. 164.

Rom Pinchasi

Let n ⩾ 1 be an odd integer. For every 1 ⩽ i ⩽ n let s_i = (a_i, b_i) be an open unit segment on the real line. Let 0 ≤ ε < ½ be fixed. Color by green all the points (numbers) on the real line of the form a_i + ε and b_i − ε. Then there exists at least one green point that belongs to an odd number of the segments s₁, …, s_n.

DOI: 10.1080/00029890.2018.1401833

A Functional Equation Characterization of Archimedean Ordered Fields

p. 169.

Ralph Howard, Virginia Johnson & George F. McNulty

We prove that an ordered field is Archimedean if and only if every continuous additive function from the field to itself is linear over the field.

DOI: 10.1080/00029890.2018.1401877

An Inequality Involving sin(n)

p. 173.

Christopher Stuart

We prove an inequality involving sine by using an estimate of the irrationality measure of π.

DOI: 10.1080/00029890.2018.1401879

Partial Fractions and Riesz’ Interpolation Formula for Trigonometric Polynomials

p. 175.

Wenchang Chu

By means of the partial fraction decomposition method, we present a new and elementary proof of the Riesz interpolation formula for trigonometric polynomials, including another interpolation formula as a by-product.

DOI: 10.1080/00029890.2018.1401854

Problems and Solutions

p. 179.

DOI: 10.1080/00029890.2017.1405685

Book Review

p. 188.

A Beginner's Guide to Mathematical Logic by Raymond M. Smullyan

Reviewed by Robert Cowen

DOI: 10.1080/00029890.2018.1401883

MathBits

Lorentz Stöer's Geometria et Perspectiva

p. 157.

100 Years Ago This Month in The American Mathematical Monthly

p. 172.