arXiv subjects

William Craig

Publications and source records attributed to William Craig.

At least 37 records · Page 2Linked to original sources

arXiv2023

Hook length biases and general linear partition inequalities

Motivated in part by hook-content formulas for certain restricted partitions in representation theory, we consider the total number of hooks of fixed length in odd versus distinct partitions. We show that there are more hooks of length $2$, respectively $3$, in all odd partitions of $n$ than in all distinct partitions of $n$, and make the analogous conjecture for arbitrary hook length $t \geq 2$. We also establish additional bias results on the number of gaps of size $1,$ respectively $2$, in all odd versus distinct partitions of $n$. We conjecture similar biases and asymptotics, as well as congruences for the number of hooks of fixed length in odd distinct partitions versus self-conjugate partitions. An integral component of the proof of our bias result for hooks of length $3$ is a linear inequality involving $q(n)$, the number of distinct partitions of $n$. In this article we also establish effective linear inequalities for $q(n)$ in great generality, a result which is of independent interest. Our methods are both analytic and combinatorial, and our results and conjectures intersect the areas of representation theory, analytic number theory, partition theory, and $q$-series. In particular, we use a Rademacher-type exact formula for $q(n),$ Wright's circle method, modularity, $q$-series transformations, asymptotic methods, and combinatorial arguments.

math.CO↗

arXiv2023

On the distribution of the norm of partitions

The norm of an integer partition is defined as the product of its parts. This statistic was recently introduced by Schneider in connection to partition zeta functions. In this note, we use the method of moments to study the distribution of the norm under the uniform probability measure on partitions of $n$ as $n \to \infty$. We use singularity analysis to prove asymptotics for the moments and show as a result that the norm lacks a non-trivial limiting distribution on $[0,\infty)$.

math.CO↗

arXiv2023

The Riemann Hypothesis for period polynomials of cusp forms

We consider the period polynomials $r_f(z)$ associated with cusp forms $f$ of weight $k$ on all of $\mathrm{SL}_2\left( \mathbb{Z} \right)$, which are generating functions for the critical $L$-values of the modular $L$-function associated to $f$. In 2014, El-Guindy and Raji proved that if $f$ is an eigenform, then $r_f(z)$ satisfies a ``Riemann hypothesis" in the sense that all its zeros lie on the natural boundary of its functional equation. We show that this phenomenon is not restricted to eigenforms, and we provide large natural infinite families of cusp forms whose period polynomials almost always satisfy the Riemann hypothesis. For example, we show that for weights $k \geq 120$, linear combinations of eigenforms with positive coefficients always have unimodular period polynomials.

math.NT↗

arXiv2022

Seaweed Algebras and the Index Statistic for Partitions

In 2018 Coll, Mayers, and Mayers conjectured that the $q$-series $( q, -q^3; q^4 )_\infty^{-1}$ is the generating function for a certain parity statistic related to the index of seaweed algebras. We prove this conjecture. Thanks to earlier work by Seo and Yee, the conjecture would follow from the non-negativity of the coefficients of this infinite product. Using a variant of the circle method along with Euler-Maclaurin summation, we establish this non-negativity, thereby confirming the Coll-Mayers-Mayers Conjecture.

math.CO↗

arXiv2022

On the Number of Parts in Congruence Classes for Partitions into Distinct Parts

For integers $0 < r \leq t$, let the function $D_{r,t}(n)$ denote the number of parts among all partitions of $n$ into distinct parts that are congruent to $r$ modulo $t$. We prove the asymptotic formula $$D_{r,t}(n) \sim \dfrac{3^{\frac 14} e^{π\sqrt{\frac{n}{3}}}}{2πt n^{\frac 14}} \left( \log(2) + \left( \dfrac{\sqrt{3} \log(2)}{8π} - \dfracπ{4\sqrt{3}} \left( r - \dfrac{t}{2} \right) \right) n^{- \frac 12} \right)$$ as $n \to \infty$. A corollary of this result is that for $0 < r < s \leq t$, the inequality $D_{r,t}(n) \geq D_{s,t}(n)$ holds for all sufficiently large $n$. We make this effective, showing that for $2 \leq t \leq 10$ the inequality $D_{r,t}(n) \geq D_{s,t}(n)$ holds for all $n > 8$.

math.NT↗

arXiv2022

On Ramanujan-type Congruences for Multiplicative Functions

The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we investigate the existence and classification of Ramanujan-type congruences for functions in multiplicative number theory.

math.NT↗

arXiv2022

Cubefree Trinomial Discriminants

The discriminant of a polynomial of the form $\pm x^n \pm x^m \pm 1$ has the form $n^n \pm m^m(n-m)^{n-m}$ when $n,m$ are relatively prime. We investigate when these discriminants have prime power divisors. We explain several symmetries that appear in the classification of these values of $n,m$. We prove that there are infinitely many pairs of integers $n,m$ for which this discriminant has no prime cube divisors. This result is extended to show that for infinitely many fixed $m$, there are infinitely many $n$ for which the discriminant has no prime cube divisor.

math.NT↗

arXiv2022

Distributions on partitions arising from Hilbert schemes and hook lengths

Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory, and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations which arise from extensions of the Nekrasov-Okounkov hook product formula, and from Betti numbers of various Hilbert schemes of $n$ points on $\mathbb{C}^2.$ For the Hilbert schemes, we prove that homology is equidistributed as $n\to \infty.$ For $t$-hooks, we prove distributions which are often not equidistributed. The cases where $t\in \{2, 3\}$ stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results which are of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products $$ F_1(ξ; q):=\prod_{n=1}^{\infty}\left(1-ξq^n\right), \ \ \ F_2(ξ; q):=\prod_{n=1}^{\infty}\left(1-(ξq)^n\right) \ \ \ {\text {and}}\ \ \ F_3(ξ; q):=\prod_{n=1}^{\infty}\left(1-ξ^{-1}(ξq)^n\right). $$

math.NT↗

arXiv2021

Asymptotic equidistribution for partition statistics and topological invariants

We provide a general framework for proving asymptotic equidistribution, convexity, and log concavity of coefficients of generating functions on arithmetic progressions. Our central tool is a variant of Wright's Circle Method proved by two of the authors with Bringmann and Ono, following work of Ngo and Rhoades. We offer a selection of different examples of such results, proving asymptotic equidistribution results for several partition statistics, modular sums of Betti numbers of two- and three-flag Hilbert schemes, and the number of cells of dimension a (mod b) of a certain scheme central in work of Göttsche.

math.NT↗

arXiv2021

A note on the Higher order Turán inequalities for $k$-regular partitions

Nicolas and DeSalvo and Pak proved that the partition function $p(n)$ is log concave for $n \geq 25$. Chen, Jia and Wang proved that $p(n)$ satisfies the third order Turán inequality, and that the associated degree 3 Jensen polynomials are hyperbolic for $n \geq 94$. More recently, Griffin, Ono, Rolen and Zagier proved more generally that for all $d$, the degree $d$ Jensen polynomials associated to $p(n)$ are hyperbolic for sufficiently large $n$. In this paper, we prove that the same result holds for the $k$-regular partition function $p_k(n)$ for $k \geq 2$. In particular, for any positive integers $d$ and $k$, the order $d$ Turán inequalities hold for $p_k(n)$ for sufficiently large $n$. The case when $d = k = 2$ proves a conjecture by Neil Sloane that $p_2(n)$ is log concave.

math.NT↗

arXiv2021

Distribution properties for t-hooks in partitions

Partitions, the partition function $p(n)$, and the hook lengths of their Ferrers-Young diagrams are important objects in combinatorics, number theory and representation theory. For positive integers $n$ and $t$, we study $p_t^e(n)$ (resp. $p_t^o(n)$), the number of partitions of $n$ with an even (resp. odd) number of $t$-hooks. We study the limiting behavior of the ratio $p_t^e(n)/p(n)$, which also gives $p_t^o(n)/p(n)$ since $p_t^e(n) + p_t^0(n) = p(n)$. For even $t$, we show that $$\lim\limits_{n \to \infty} \dfrac{p_t^e(n)}{p(n)} = \dfrac{1}{2},$$ and for odd $t$ we establish the non-uniform distribution $$\lim\limits_{n \to \infty} \dfrac{p^e_t(n)}{p(n)} = \begin{cases} \dfrac{1}{2} + \dfrac{1}{2^{(t+1)/2}} & \text{if } 2 \mid n, \\ \\ \dfrac{1}{2} - \dfrac{1}{2^{(t+1)/2}} & \text{otherwise.} \end{cases}$$ Using the Rademacher circle method, we find an exact formula for $p_t^e(n)$ and $p_t^o(n)$, and this exact formula yields these distribution properties for large $n$. We also show that for sufficiently large $n$, the signs of $p_t^e(n) - p_t^o(n)$ are periodic.

math.NT↗

arXiv2020

Unique product groups and congruence subgroups

We prove that a uniform pro-p group with no nonabelian free subgroups has a normal series with torsion-free abelian factors. We discuss this in relation to unique product groups. We also consider generalizations of Hantzsche-Wendt groups.

math.GR↗

arXiv2020

Variations of Lehmer's Conjecture for Ramanujan's tau-function

We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for $n>1$ we prove that $$τ(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 691\}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujan's well-known congruences for $τ(n)$ allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty-Coleman method for hyperelliptic curves, and facts about certain Thue equations.

math.NT↗

arXiv2018

An Indirect Dark Matter Search Using Cosmic-Ray Antiparticles with GAPS

Experiments aiming to directly detect dark matter (DM) particles have yet to make robust detections, thus underscoring the need for complementary approaches such as searches for new particles at colliders, and indirect DM searches in cosmic-ray spectra. Low energy (< 0.25 GeV/n) cosmic-ray antiparticles such as antideuterons are strong candidates for probing DM models, as the yield of these particles from DM processes can exceed the astrophysical background by more than two orders of magnitude. The General Antiparticle Spectrometer (GAPS), a balloon borne cosmic-ray detector, will perform an ultra-low background measurement of the cosmic antideuteron flux in the regime < 0.25 GeV/n, which will constrain a wide range of DM models. GAPS will also detect approximately 1000 antiprotons in an unexplored energy range throughout one long duration balloon (LDB) flight, which will constrain < 10 GeV DM models and validate the GAPS detection technique. Unlike magnetic spectrometers, GAPS relies on the formation of an exotic atom within the tracker in order to identify antiparticles. The GAPS tracker consists of ten layers of lithium-drifted silicon detectors which record dE/dx deposits from primary and nuclear annihilation product tracks, as well as measure the energy of the exotic atom deexcitation X-rays. A two-layer, plastic scintillator time of flight (TOF) system surrounds the tracker and measures the particle velocity, dE/dx deposits, and provides a fast trigger to the tracker. The nuclear annihilation product multiplicity, deexcitation X-ray energies, TOF, and stopping depth are all used together to discern between antiparticle species. This presentation provided an overview of the GAPS experiment, an update on the construction of the tracker and TOF systems, and a summary of the expected performance of GAPS in light of the upcoming LDB flight from McMurdo Station, Antarctica in 2020.

physics.ins-det↗

arXiv2017

The X-ray reflection spectrum of the radio-loud quasar 4C 74.26

The relativistic jets created by some active galactic nuclei are important agents of AGN feedback. In spite of this, our understanding of what produces these jets is still incomplete. X-ray observations, which can probe the processes operating in the central regions in immediate vicinity of the supermassive black hole, the presumed jet launching point, are potentially particularly valuable in illuminating the jet formation process. Here, we present the hard X-ray NuSTAR observations of the radio-loud quasar 4C 74.26 in a joint analysis with quasi-simultaneous, soft X-ray Swift observations. Our spectral analysis reveals a high-energy cut-off of 183$_{-35}^{+51}$ keV and confirms the presence of ionized reflection in the source. From the average spectrum we detect that the accretion disk is mildly recessed with an inner radius of $R_\mathrm{in}=4-180\,R_\mathrm{g}$. However, no significant evolution of the inner radius is seen during the three months covered by our NuSTAR campaign. This lack of variation could mean that the jet formation in this radio-loud quasar differs from what is observed in broad-line radio galaxies.

astro-ph.HE↗

arXiv2016

The Geometry of the Infrared and X-ray Obscurer in a Dusty Hyperluminous Quasar

We study the geometry of the AGN obscurer in IRAS 09104+4109, an IR-luminous, radio-intermediate FR-I source at $z=0.442$, using infrared data from Spitzer and Herschel, X-ray data from Nustar, Swift, Suzaku, and Chandra, and an optical spectrum from Palomar. The infrared data imply a total rest-frame 1-1000$μ$m luminosity of $5.5\times10^{46}$erg s$^{-1}$ and require both an AGN torus and starburst model. The AGN torus has an anisotropy-corrected IR luminosity of $4.9\times10^{46}$erg s$^{-1}$, and a viewing angle and half opening angle both of approximately $36$ degrees from pole-on. The starburst has a star formation rate of $(110\pm34)$M$_{\odot}$ yr$^{-1}$ and an age of $<50$Myr. These results are consistent with two epochs of luminous activity in 09104: one approximately $150$Myr ago, and one ongoing. The X-ray data suggest a photon index of $Γ\simeq 1.8$ and a line-of-sight column of $N_{\rm H} \simeq 5\times10^{23}$cm$^{-2}$. This argues against a reflection-dominated hard X-ray spectrum, which would have implied a much higher $N_{\rm H}$ and luminosity. The X-ray and infrared data are consistent with a bolometric AGN luminosity of $L_{\rm bol}\sim(0.5-2.5)\times10^{47}$erg s$^{-1}$. The X-ray and infrared data are further consistent with coaligned AGN obscurers in which the line of sight "skims" the torus. This is also consistent with the optical spectra, which show both coronal iron lines and broad lines in polarized but not direct light. Combining constraints from the X-ray, optical, and infrared data suggests that the AGN obscurer is within a vertical height of $20$pc, and a radius of $125$pc, of the nucleus.

astro-ph.GA↗

arXiv2016

Characterizing X-ray and Radio emission in the Black Hole X-Ray Binary V404 Cygni during Quiescence

We present results from multi-wavelength simultaneous X-ray and radio observations of the black hole X-ray binary V404 Cyg in quiescence. Our coverage with NuSTAR provides the very first opportunity to study the X-ray spectrum of V404 Cyg at energies above 10 keV. The unabsorbed broad-band (0.3--30 keV) quiescent luminosity of the source is 8.9$\times$10$^{32}$ erg s$^{-1}$ for a distance of 2.4 kpc. The source shows clear variability on short time scales (an hour to a couple of hours) in radio, soft X-ray and hard X-ray bands in the form of multiple flares. The broad-band X-ray spectra obtained from XMM-Newton and NuSTAR can be characterized with a power-law model having photon index $Γ$=2.12$\pm$0.07 (90\% confidence errors); however, residuals at high energies indicate spectral curvature significant at a 3$σ$ confidence level with e-folding energy of the cutoff to be 20$^{+20}_{-7}$ keV. Such curvature can be explained using synchrotron emission from the base of a jet outflow. Radio observations using the VLA reveal that the spectral index evolves on very fast time-scales (as short as 10 min.), switching between optically thick and thin synchrotron emission, possibly due to instabilities in the compact jet or stochastic instabilities in accretion rate. We explore different scenarios to explain this very fast variability.

astro-ph.HE↗

arXiv2016

The rhythm of Fairall 9 -- I: Observing the spectral variability with XMM-Newton and NuSTAR

We present a multi-epoch X-ray spectral analysis of the Seyfert 1 galaxy Fairall 9. Our analysis shows that Fairall 9 displays unique spectral variability in that its ratio residuals to a simple absorbed power law in the 0.5-10 keV band remain constant with time in spite of large variations in flux. This behavior implies an unchanging source geometry and the same emission processes continuously at work at the timescale probed. With the constraints from NuSTAR on the broad-band spectral shape, it is clear that the soft excess in this source is a superposition of two different processes, one being blurred ionized reflection in the innermost parts of the accretion disk, and the other a continuum component such as spatially distinct Comptonizing region. Alternatively, a more complex primary Comptonization component together with blurred ionized reflection could be responsible.

astro-ph.GA↗