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Cormac O'Sullivan

Publications and source records attributed to Cormac O'Sullivan.

34 records · Page 2Linked to original sources
arXiv2018

Revisiting the saddle-point method of Perron

Perron's saddle-point method gives a way to find the complete asymptotic expansion of certain integrals that depend on a parameter going to infinity. We give two proofs of the key result. The first is a reworking of Perron's original proof, showing the clarity and simplicity that has been lost in some subsequent treatments. The second proof extends the approach of Olver which is based on Laplace's method. New results include more precise error terms and bounds for the expansion coefficients. We also treat Perron's original examples in greater detail and give a new application to the asymptotics of Sylvester waves.

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arXiv2017

Hyperbolic Fourier coefficients of Poincaré series

Poincaré in 1911 and Petersson in 1932 gave the now classical expression for the parabolic Fourier coefficients of holomorphic Poincaré series in terms of Bessel functions and Kloosterman sums. Later, in 1941, Petersson introduced hyperbolic and elliptic Fourier expansions of modular forms and the associated hyperbolic and elliptic Poincaré series. In this paper we express the hyperbolic Fourier coefficients of Poincaré series, of both parabolic and hyperbolic type, in terms of hypergeometric series and Good's generalized Kloosterman sums. In an explicit example for the modular group, we see that the hyperbolic Kloosterman sum corresponds to a sum over lattice points on a hyperbola contained in an ellipse. This allows for numerical computation of the hyperbolic Fourier coefficients.

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arXiv2016

Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series

Let $E(z,s)$ be the non-holomorphic Eisenstein series for the modular group $SL(2,{\mathbb Z})$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $s=1$ of $E(z,s)$ is essentially the logarithm of the Dedekind eta function. This eta function is a weight $1/2$ modular form and Dedekind expressed its multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups $Γ$. The analogue of the eta function has a multiplier system that may be expressed in terms of a map $S:Γ\to {\mathbb R}$ which we call a modular Dedekind symbol. We obtain detailed properties of these symbols by means of the limit formula. Twisting the usual Eisenstein series with powers of additive homomorphisms from $Γ$ to ${\mathbb C}$ produces higher-order Eisenstein series. These series share many of the properties of $E(z,s)$ though they have a more complicated automorphy condition. They satisfy a Kronecker limit formula and produce higher-order Dedekind symbols $S^*:Γ\to {\mathbb R}$. As an application of our general results, we prove that higher-order Dedekind symbols associated to genus one congruence groups $Γ_0(N)$ are rational.

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arXiv2015

Asymptotics for the partial fractions of the restricted partition generating function I

The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. We find the behavior of coefficients in the partial fraction decomposition of this product as $N \to \infty$ by applying the saddle-point method, where the saddle-point we need is associated to a zero of the analytically continued dilogarithm. Our main result disproves a conjecture of Rademacher.

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arXiv2015

Asymptotics for the partial fractions of the restricted partition generating function II

The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. In part I, we studied the behavior of coefficients in the partial fraction decomposition of this product as $N \to \infty$ by applying the saddle-point method to get the asymptotics of the main terms. In this second part we bound the error terms. This involves estimating products of sines and further saddle-point arguments. The saddle-points needed are associated to zeros of the analytically continued dilogarithm.

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arXiv2015

Zeros of the dilogarithm

We show that the dilogarithm has at most one zero on each branch, that each zero is close to a root of unity, and that they may be found to any precision with Newton's method. This work is motivated by applications to the asymptotics of coefficients in partial fraction decompositions considered by Rademacher. We also survey what is known about zeros of polylogarithms in general.

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arXiv2014

On the partial fraction decomposition of the restricted partition generating function

We provide new formulas for the coefficients in the partial fraction decomposition of the restricted partition generating function. These techniques allow us to partially resolve a recent conjecture of Sills and Zeilberger. We also describe upcoming work, giving a resolution to Rademacher's conjecture on the asymptotics of these coefficients.

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arXiv2012

Non-vanishing of Taylor coefficients and Poincaré series

We prove recursive formulas for the Taylor coefficients of cusp forms, such as Ramanujan's Delta function, at points in the upper half-plane. This allows us to show the non-vanishing of all Taylor coefficients of Delta at CM points of small discriminant as well as the non-vanishing of certain Poincaré series. At a "generic" point all Taylor coefficients are shown to be non-zero. Some conjectures on the Taylor coefficients of Delta at CM points are stated.

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arXiv2011

Kernels for products of L-functions

The Rankin-Cohen bracket of two Eisenstein series provides a kernel yielding products of the periods of Hecke eigenforms at critical values. Extending this idea leads to a new type of Eisenstein series built with a double sum. We develop the properties of these series and their non-holomorphic analogs and show their connection to values of L-functions outside the critical strip.

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arXiv2009

A Diophantine Frobenius problem related to Riemann surfaces

We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair $(p,q)$, $2<p<q$, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realized as a semi-regular $pq$-fold coverings of surfaces of lower genus. In this context, the Frobenius number is (up to an additive translation) the largest genus in which no surface is such a covering. In many cases it is also the largest genus in which no surface admits an automorphism of order $pq$. The general $t$-dimensional Frobenius problem ($t \geq 3$) is $NP$-hard, and it may be that our restricted problem retains this property.

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arXiv2009

Kernels of $L$-functions of cusp forms

We give a new expression for the inner product of two kernel functions associated to a cusp form. Among other applications, it yields an extension of a formula of Kohnen and Zagier, and another proof of Manin's Periods Theorem. Cohen's representation of these kernels as series is also generalized.

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arXiv2008

Parabolic, hyperbolic and elliptic Poincaré series

Following Petersson, we study the parabolic, hyperbolic and elliptic expansions of holomorphic cusp forms and the associated Poincaré series. We show how these ideas extend to the space of second-order cusp forms.

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arXiv2007

On the Linear Independence of Roots

A set of real $n$th roots that is pairwise linearly independent over the rationals must also be linearly independent. We show how this result may be extended to more general fields.

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arXiv2006

Unipotent vector bundles and higher-order non-holomorphic Eisenstein series

Higher-order non-holomorphic Eisenstein series associated to a Fuchsian group $Γ$ are defined by twisting the series expansion for classical non-holomorphic Eisenstein series by powers of modular symbols. Their functional identities include multiplicative and additive factors, making them distinct from classical Eisenstein series. Here we prove the meromorphic continuation of these series and establish their functional equations. In addition, we construct high rank vector bundles $\cal V$ from certain unipotent representations $π$ of $Γ$ and show that higher-order non-holomorphic Eisenstein series can be viewed as components of certain eigensections, $\mathbb E$, of $\cal V$. With this viewpoint the functional identities of these higher-order series are formally identical to the classical case. Going further, we prove bounds for the Fourier coefficients of the higher-order non-holomorphic Eisenstein series.

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