The 10 most-cited articles in American Mathematical Monthly, 2018

In this note, we prove an improvement of the classical discrete Hardy inequality. Our improved Hardy-type inequality holds with a weight w which is strictly greater than the classical Hardy weight wH(n) ≔ 1/(2n)2, where .

Since its inception in 1894, the Monthly has printed 50 articles on the Γ function or Stirling's asymptotic formula, including the magisterial 1959 paper by Phillip J. Davis, which won the 1963 Chauvenet prize, and the eye-opening 2000 paper by the Fields medalist Manjul Bhargava. In this article, we look back and comment on what has been said, and why, and try to guess what will be said about the Γ function in future Monthly issues.11 It is a safe bet that there will be more proofs of Stirling's formula, for instance. We also identify some gaps, which surprised us: phase plots, Riemann surfaces, and the functional inverse of Γ make their first appearance in the Monthly here. We also give a new elementary treatment of the asymptotics of n! and the first few terms of a new asymptotic formula for invΓ.

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.

In 2018, the U.S. Supreme Court considered a proposed mathematical formula to help detect unconstitutional partisan gerrymandering. We show that in some cases, this formula only flags bizarrely-shaped districts as potentially constitutional.

For a fixed we say that a point (r, s) in the integer lattice is b-visible from the origin if it lies on the graph of a power function f(x) = axb with and no other integer lattice point lies on this curve (i.e., line of sight) between (0, 0) and (r, s). We prove that the proportion of b-visible integer lattice points is given by 1/ζ(b + 1), where ζ(s) denotes the Riemann zeta function. We also show that even though the proportion of b-visible lattice points approaches 1 as b approaches infinity, there exist arbitrarily large rectangular arrays of b-invisible lattice points for any fixed b. This work specialized to b = 1 recovers original results from the classical lattice point visibility setting where the lines of sight are given by linear functions with rational slope through the origin.

Service sports include two-player contests such as volleyball, badminton, and squash. We analyze four rules, including the Standard Rule (SR), in which a player continues to serve until he or she loses. The Catch-Up Rule (CR) gives the serve to the player who has lost the previous point—as opposed to the player who won the previous point, as under SR. We also consider two Trailing Rules that make the server the player who trails in total score. Surprisingly, compared with SR, only CR gives the players the same probability of winning a game while increasing its expected length, thereby making it more competitive and exciting to watch. Unlike one of the Trailing Rules, CR is strategy-proof. By contrast, the rules of tennis fix who serves and when; its tiebreaker, however, keeps play competitive by being fair—not favoring either the player who serves first or who serves second.

We construct arbitrarily large sets of dice with some remarkable nontransitivity properties. In a sense made precise later, each set exhibits all possible pairwise win/loss relationships by summing different numbers of rolls. The proof of this fact relies on an asymptotic formula for the difference between the median and mean of sums of multiple rolls of dice. This formula is a consequence of a suitable Edgeworth series (an asymptotic refinement of the central limit theorem), for which we give a detailed sketch of a proof in the final section.

In this article, we use elementary methods to investigate continuous binomial coefficients: functions of the real variable x defined by way of the gamma function with y a fixed real number. We begin with a brief qualitative description of these functions and then derive several interesting representations of them including an infinite product and Taylor series. We also prove various integral formulas involving continuous binomial coefficients, many of which remarkably mirror summation formulas of the familiar binomial coefficients. We conclude by proving a continuous analog of the binomial theorem.

In this note, we use the Lambert series generating function for Euler’s totient function to introduce a new identity for the number of 1’s in the partitions of n. A new expansion for Euler’s partition function p(n) is derived in this context. These surprising new results connect the famous classical totient function from multiplicative number theory to the additive theory of partitions.

One of the strangest, most mind-boggling examples in analysis is that of a function from to that is everywhere differentiable but monotone on no interval. The graph of such a “monstrous” function is simultaneously smooth and very rugged. Although such examples have been known for over 100 years, so far the existing constructions are quite involved. In this note, we provide a simple example of such a map. It is presented in a broader context of other paradoxical examples related to differentiability of continuous maps from to , including a differentiable function that maps a compact perfect subset of onto itself even though its derivative vanishes on .