arXiv subjects

Marc Chamberland

Publications and source records attributed to Marc Chamberland.

7 recordsLinked to original sources

arXiv2026

Weakening the Legendre Conjecture

The world of primes has many gaps between evidence and theorems. Here, we review Legendre's conjecture on primes between consecutive squares and recent progress on the weaker question of primes between consecutive larger powers. Assuming the Riemann hypothesis (RH), we observe that a recent result of Emanuel Carneiro, Micah Milinovich and Kannan Soundararajan, combined with a large-scale computation by Jonathan Sorenson and Jonathan Webster, implies the existence of primes between $x^{2+\delta}$ and $(x+1)^{2+\delta}$ for all real $x \geq 1$ when $\delta \geq 1/4$. For smaller values of $\delta > 0$, we provide an explicit bound $x_0 = x_0 (\delta)$ such that primes exist in these intervals whenever $x \geq x_0$ (again assuming RH). We conclude with an application to Mills-type prime-generating constants.

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arXiv2025

An alternating sum of the floor function of square roots

We show that the alternating sum of the floor function of $\sqrt{jn}$, with $j$ ranging from 1 to $n$, has an easy evaluation for all odd integers $n\geq 1$. This is in contrast to known non-alternating sums of the same type which hold only for a class of primes. The proof is elementary and was suggested by an AI model. To put this result in perspective, we also prove an asymptotic expression for the analogous sum without the floor function.

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arXiv2025

Sums of the floor function related to class numbers of imaginary quadratic fields

A curious identity of Bunyakovsky (1882), made more widely known by P\'olya and Szeg{\H o} in their ``Problems and Theorems in Analysis", gives an evaluation of a sum of the floor function of square roots involving primes $p\equiv 1\pmod{4}$. We evaluate this sum also in the case $p\equiv 3\pmod{4}$, obtaining an identity in terms of the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-p})$. We also consider certain cases where the prime $p$ is replaced by a composite integer. Class numbers of imaginary quadratic fields are again involved in some cases.

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arXiv2024

Formulas for odd zeta values and powers of $π$

Plouffe conjectured rapidly converging series formulas for $π^{2n+1}$ and $ζ(2n+1)$ for small values of $n$. We find the general pattern for all nonnegative integer values of $n$ and offer a proof.

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arXiv2020

Apéry Limits: Experiments and Proofs

An important component of Apéry's proof that $ζ(3)$ is irrational involves representing $ζ(3)$ as the limit of the quotient of two rational solutions to a three-term recurrence. We present various approaches to such Apéry limits and highlight connections to continued fractions as well as the famous theorems of Poincaré and Perron on difference equations. In the spirit of Jon Borwein, we advertise an experimental-mathematics approach by first exploring in detail a simple but instructive motivating example. We conclude with various open problems.

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arXiv2013

On gamma quotients and infinite products

Convergent infinite products, indexed by all natural numbers, in which each factor is a rational function of the index, can always be evaluated in terms of finite products of gamma functions. This goes back to Euler. A purpose of this note is to demonstrate the usefulness of this fact through a number of diverse applications involving multiplicative partitions, entries in Ramanujan's notebooks, the Chowla--Selberg formula, and the Thue--Morse sequence. In addition, we propose a numerical method for efficiently evaluating more general infinite series such as the slowly convergent Kepler--Bouwkamp constant.

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arXiv2013

A Short Proof of a Ptolemy-Like Relation for an Even number of Points on a Circle Discovered by Jane McDougall

We give a short proof of a Ptolemy-style result first discovered and proved by Jane McDougall. It may be viewed as a generalization to any even number of points of the cubic relation connecting the six joint distances of four points on a circle (whose quadratic relation is the famed Plucker relation, alias Ptolemy's theorem). This article is in fond memory of Andrei Zelevinsky (1953-2013) who loved Ptolemy's theorem.

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