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Cormac O'Sullivan

Publications and source records attributed to Cormac O'Sullivan.

At least 19 recordsLinked to original sources
arXiv2026

Patterns in the Markov numbers and their generalizations

Positive integer solutions to $x^2+y^2+z^2- x y z =D$ correspond to the important Markov (Markoff) numbers when $D=0$. From a given solution triple, three more are found with Vieta involutions, making an infinite tree of solutions. Starting instead with three real numbers greater than $2$ gives lengths of closed geodesics in a punctured torus. In this paper we study these tree structures for any real $D$. A continuous function, originally related to a norm on homology, encodes all the numbers on each of these trees. The properties of this function are developed here in general, with a self-contained exposition, showing that the usual $D=0$ case is part of a bigger picture. Encoding function graphs are shown to be convex for $D<4$, straight lines for $D=4$ and concave for $D>4$. Among other consequences are generalizations to all $D$ of: estimates for counting numbers on these trees, descriptions of the geometry of the corresponding lattice curves, uniqueness conditions, and identities of McShane and Hines.

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arXiv2025

Integer continued fractions for complex numbers

We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new representations are shown to be unique, and to have useful properties. They also admit a geometric cutting sequence interpretation.

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arXiv2025

Topographs for binary quadratic forms and class numbers

In this work we study, in greater detail than before, J.H. Conway's topographs for integral binary quadratic forms. These are trees in the plane with regions labeled by integers following a simple pattern. Each topograph can display the values of a single form, or represent an equivalence class of forms. We give a new treatment of reduction of forms to canonical equivalence class representatives by employing topographs and a novel continued fraction for complex numbers. This allows uniform reduction for any positive, negative, square or non-square discriminant. Topograph geometry also provides new class number formulas, and short proofs of results of Gauss relating to sums of three squares. Generalizations of the series of Hurwitz for class numbers give evaluations of certain infinite series, summed over the regions or edges of a topograph.

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arXiv2023

Integrality of the higher Rademacher symbols

Rademacher symbols may be defined in terms of Dedekind sums, and give the value at zero of the zeta function associated to a narrow ideal class of a real quadratic field. Duke extended these symbols to give the zeta function values at all negative integers. Here we prove Duke's conjecture that these higher Rademacher symbols are integer valued, making the above zeta value denominators as simple as the corresponding Riemann zeta value denominators. The proof uses detailed properties of Bernoulli numbers, including a generalization of the Kummer congruences.

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arXiv2023

Detailed asymptotic expansions for partitions into powers

Here we examine the number of ways to partition an integer $n$ into $k$th powers when $n$ is large. Simplified proofs of some asymptotic results of Wright are given using the saddle-point method, including exact formulas for the expansion coefficients. The convexity and log-concavity of these partitions is shown for large $n$, and the stronger conjectures of Ulas are proved. The asymptotics of Wright's generalized Bessel functions are also treated.

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arXiv2023

Stirling's approximation and a hidden link between two of Ramanujan's approximations

A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of these, and Stirling's approximation to the gamma function. Our work provides new information about the coefficients in Stirling's approximation and their connection to Ramanujan's approximation coefficients. A more analytic second proof of the main result is also included in an appendix.

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arXiv2022

Ramanujan's approximation to the exponential function and generalizations

Ramanujan's approximation to the exponential function is reexamined with the help of Perron's saddle-point method. This allows for a wide generalization that includes the results of Buckholtz, and where all the asymptotic expansion coefficients may be given in closed form. Ramanujan's approximation to the exponential integral is treated similarly.

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arXiv2022

De Moivre and Bell polynomials

We survey a family of polynomials that are very useful in all kinds of power series manipulations, and appearing more frequently in the literature. Applications to formal power series, generating functions and asymptotic expansions are described, and we discuss the related work of De Moivre, Arbogast and Bell.

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arXiv2022

Symmetric functions and a natural framework for combinatorial and number theoretic sequences

Certain triples of power series, considered by I. Macdonald, give a natural framework for many combinatorial and number theoretic sequences, such as the Stirling, Bernoulli and harmonic numbers and partitions of different kinds. The power series in such a triple are closely linked by identities coming from the theory of symmetric functions. We extend the work of Z-H. Sun, who developed similar ideas, and Macdonald, revealing more of the structure of these triples. De Moivre polynomials play a key role in this study.

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arXiv2021

Limits of Jensen polynomials for partitions and other sequences

It was discovered recently by Griffin, Ono, Rolen and Zagier that the Jensen polynomials associated to many sequences have Hermite polynomial limits. We develop this theory in detail, based on the log-polynomial property which is a refinement of log-concavity and log-convexity. Applications to various partition sequences are given. An application to the sequence of factorials leads naturally to evaluating limits of generalized Laguerre polynomials.

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arXiv2020

Zeros of Jensen polynomials and asymptotics for the Riemann xi function

The classical criterion of Jensen for the Riemann hypothesis is that all of the associated Jensen polynomials have only real zeros. We find a new version of this criterion, using linear combinations of Hermite polynomials, and show that this condition holds in many cases. Detailed asymptotic expansions are given for the required Taylor coefficients of the xi function at $1/2$ as well as related quantities. These results build on those in the recent paper of Griffin, Ono, Rolen and Zagier.

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arXiv2020

Rademacher's conjecture and expansions at roots of unity of products generating restricted partitions

The generating function for restricted partitions is a finite product with a Laurent expansion at each root of unity. The question of the behavior of these Laurent coefficients as the size of the product increases goes back to Rademacher and his work on partitions. Building on the methods of Drmota, Gerhold and previous results of the author, we complete this description and give the full asymptotic expansion of each coefficient at every root of unity. These techniques are also shown to give the asymptotics of Sylvester waves.

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arXiv2019

Transformation laws for generalized Dedekind sums associated to Fuchsian groups

We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kind, and examples are provided in the cases of the Hecke triangle groups, the Hecke congruence groups $Γ_0(N)$, and the non-congruence arithmetic groups $Γ_0(N)^+$.

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arXiv2019

A generalization of the Riemann-Siegel formula

The celebrated Riemann-Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable $t$. Siegel anticipated that this formula could be generalized to include the Hardy-Littlewood approximate functional equation, valid in any vertical strip. We give this generalization for the first time. The asymptotics contain Mordell integrals and an interesting new family of polynomials.

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arXiv2018

Noncommutative modular symbols and Eisenstein series

We form real-analytic Eisenstein series twisted by Manin's noncommutative modular symbols. After developing their basic properties, these series are shown to have meromorphic continuations to the entire complex plane and satisfy functional equations in some cases. This theory neatly contains and generalizes earlier work in the literature on the properties of Eisenstein series twisted by classical modular symbols.

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arXiv2018

A vertex and edge deletion game on graphs

Starting with a graph, two players take turns in either deleting an edge or deleting a vertex and all incident edges. The player removing the last vertex wins. We review the known results for this game and extend the computation of nim-values to new families of graphs. A conjecture of Khandhawit and Ye on the nim-values of graphs with one odd cycle is proved. We also see that, for wheels and their subgraphs, this game exhibits a surprising amount of unexplained regularity.

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arXiv2018

Partitions and Sylvester waves

The restricted partition function $p_N(n)$ counts the partitions of the integer $n$ into at most $N$ parts. In the nineteenth century Sylvester described these partitions as a sum of waves. We give detailed descriptions of these waves and, for the first time, show the asymptotics of the initial waves as $N$ and $n$ both go to infinity at about the same rate. This allows us to see when the initial waves are a good approximation to $p_N(n)$ in this situation. Our proofs employ the saddle-point method of Perron and the dilogarithm.

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