arXiv subjects

Robert Schneider

Publications and source records attributed to Robert Schneider.

At least 37 records · Page 2Linked to original sources

arXiv2021

A "supernormal" partition statistic

We study a bijective map from integer partitions to the prime factorizations of integers that we call the "supernorm" of a partition, in which the multiplicities of the parts of partitions are mapped to the multiplicities of prime factors of natural numbers. The supernorm is connected to a family of maps we define, which suggests the potential to apply techniques from partition theory to identify and prove multiplicative properties of integers. We make a brief study of pertinent analytic aspects of the supernorm. Then, as an application of "supernorma"' mappings (i.e., pertaining to the supernorm statistic), we prove an analogue of a formula of Kural-McDonald-Sah to give arithmetic densities of subsets of $\mathbb N$ instead of natural densities in $\mathbb P$ like previous formulas of this type; this builds on works of Alladi, Ono, Wagner, and the first and third authors. Finally, using a table of "supernormal" additive-multiplicative correspondences, we conjecture Abelian-type formulas that specialize to our main theorem and other known results.

math.NT↗

arXiv2021

Semi-modular forms from Fibonacci-Eisenstein series

In recent work, M. Just and the second author defined a class of "semi-modular forms" on $\mathbb C$, in analogy with classical modular forms, that are "half modular" in a particular sense; and constructed families of such functions as Eisenstein-like series using symmetries related to integer partitions. Looking for further natural examples of semi-modular behavior, here we construct a family of Eisenstein-like series to produce semi-modular forms, using symmetries related to Fibonacci numbers instead of partitions. We then consider other Lucas sequences that yield semi-modular forms.

math.NT↗

arXiv2021

Partition-theoretic formulas for arithmetic densities, II

In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers $\mathbb N$ as limiting values of $q$-series as $q\to ζ$ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of $\mathbb N$ by analogous structures in the integer partitions $\mathcal P$. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of Dirichlet series arising from Dirichlet convolutions. Here the authors prove that Wang's extension has a partition-theoretic analogue as well, yielding new $q$-series density formulas for any subset of $\mathbb N$. To do so, we outline a theory of $q$-series density calculations from first principles, based on a statistic we call the "$q$-density" of a given subset. This theory in turn yields infinite families of further formulas for arithmetic densities.

math.NT↗

arXiv2021

The product of parts or "norm" of a partition

In this article we study the "norm" of an integer partition, which we define to be the product of the parts. This partition-theoretic statistic has appeared here and there in the literature of the last century or so, and is at the heart of current research by both authors. We survey known results and give new results related to this all-but-overlooked object, which, it turns out, plays a comparable role in partition theory to the size, length, and other standard partition statistics.

math.NT↗

arXiv2020

Eulerian series, zeta functions and the arithmetic of partitions

In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and analytic number theory. We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and explore new classes of partition-theoretic zeta functions and Dirichlet series -- as well as "Eulerian" $q$-hypergeometric series -- enjoying many interesting relations. We find a number of theorems of classical number theory and analysis arise as particular cases of extremely general combinatorial structure laws. Among our applications, we prove explicit formulas for the coefficients of the $q$-bracket of Bloch-Okounkov, a partition-theoretic operator from statistical physics related to quasi-modular forms; we prove partition formulas for arithmetic densities of certain subsets of the integers, giving $q$-series formulas to evaluate the Riemann zeta function; we study $q$-hypergeometric series related to quantum modular forms and the "strange" function of Kontsevich; and we show how Ramanujan's odd-order mock theta functions (and, more generally, the universal mock theta function $g_3$ of Gordon-McIntosh) arise from the reciprocal of the Jacobi triple product via the $q$-bracket operator, connecting also to unimodal sequences in combinatorics and quantum modular-like phenomena.

math.NT↗

arXiv2020

Nuclear partitions and a formula for $p(n)$

Define a "nuclear partition" to be an integer partition with no part equal to one. In this study we prove a simple formula to compute the partition function $p(n)$ by counting only the nuclear partitions of $n$, a vanishingly small subset by comparison with all partitions of $n$ as $n\to \infty$. Variations on the proof yield other formulas for $p(n)$, as well as Ramanujan-like congruences and an application to parity of the partition function.

math.NT↗

arXiv2020

Digit sums and generating functions

We connect a primitive operation from arithmetic -- summing the digits of a base-$B$ integer -- to $q$-series and product generating functions analogous to those in partition theory. We find digit sum generating functions to be intertwined with distinctive classes of "$B$-ary" Lambert series, which themselves enjoy nice transformations. We also consider digit sum Dirichlet series.

math.NT↗

arXiv2020

Sequentially congruent partitions and partitions into squares

In recent work, M. Schneider and the first author studied a curious class of integer partitions called "sequentially congruent" partitions: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the number of parts. Let $p_{\mathcal S}(n)$ be the number of sequentially congruent partitions of $n,$ and let $p_{\square}(n)$ be the number of partitions of $n$ wherein all parts are squares. In this note we prove bijectively, for all $n\geq 1,$ that $p_{\mathcal S}(n) = p_{\square}(n).$ Our proof naturally extends to show other exotic classes of partitions of $n$ are in bijection with certain partitions of $n$ into $k$th powers.

math.NT↗

arXiv2020

Sequentially congruent partitions and related bijections

We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the length of the partition. It turns out these obscure-seeming objects are embedded in a natural way in partition theory. We show that sequentially congruent partitions with largest part $n$ are in bijection with the partitions of $n$. Moreover, we show sequentially congruent partitions induce a bijection between partitions of $n$ and partitions of length $n$ whose parts obey a strict "frequency congruence" condition -- the frequency (or multiplicity) of each part is divisible by that part -- and prove families of similar bijections, connecting with G. E. Andrews's theory of partition ideals.

math.NT↗

arXiv2019

Analysis and combinatorics of partition zeta functions

We examine "partition zeta functions" analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon's partial fraction decomposition of the generating function for partitions of fixed length.

math.NT↗

arXiv2019

Phonon-assisted emission and absorption of individual color centers in hexagonal boron nitride

Defect centers in hexagonal boron nitride represent room-temperature single-photon sources in a layered van der Waals material. These light emitters appear with a wide range of transition energies ranging over the entire visible spectrum, which renders the identification of the underlying atomic structure challenging. In addition to their eminent properties as quantum light emitters, the coupling to phonons is remarkable. Their photoluminescence exhibits significant side band emission well separated from the zero phonon line (ZPL) and an asymmetric broadening of the ZPL itself. In this combined theoretical and experimental study we show that the phonon side bands can be well described in terms of the coupling to bulk longitudinal optical (LO) phonons. To describe the ZPL asymmetry we show that in addition to the coupling to longitudinal acoustic (LA) phonons also the coupling to local mode oscillations of the defect center with respect to the entire host crystal has to be considered. By studying the influence of the emitter's wave function dimensions on the phonon side bands we find reasonable values for size of the wave function and the deformation potentials. We perform photoluminescence excitation measurements to demonstrate that the excitation of the emitters is most efficient by LO-phonon assisted absorption.

cond-mat.mes-hall↗

arXiv2018

Zeeman spectroscopy of excitons and hybridization of electronic states in few-layer WSe$_2$, MoSe$_2$ and MoTe$_2$

Monolayers and multilayers of semiconducting transition metal dichalcogenides (TMDCs) offer an ideal platform to explore valley-selective physics with promising applications in valleytronics and information processing. Here we manipulate the energetic degeneracy of the $\mathrm{K}^+$ and $\mathrm{K}^-$ valleys in few-layer TMDCs. We perform high-field magneto-reflectance spectroscopy on WSe$_2$, MoSe$_2$, and MoTe$_2$ crystals of thickness from monolayer to the bulk limit under magnetic fields up to 30 T applied perpendicular to the sample plane. Because of a strong spin-layer locking, the ground state A excitons exhibit a monolayer-like valley Zeeman splitting with a negative $g$-factor, whose magnitude increases monotonically when thinning the crystal down from bulk to a monolayer. Using the $\mathbf{k\cdot p}$ calculation, we demonstrate that the observed evolution of $g$-factors for different materials is well accounted for by hybridization of electronic states in the $\mathrm{K}^+$ and $\mathrm{K}^-$ valleys. The mixing of the valence and conduction band states induced by the interlayer interaction decreases the $g$-factor magnitude with an increasing layer number. The effect is the largest for MoTe$_2$, followed by MoSe$_2$, and smallest for WSe$_2$. Keywords: MoSe$_2$, WSe$_2$, MoTe$_2$, valley Zeeman splitting, transition metal dichalcogenides, excitons, magneto optics.

cond-mat.mes-hall↗

arXiv2017

Jacobi's triple product, mock theta functions, unimodal sequences and the $q$-bracket

In Ramanujan's final letter to Hardy, he listed examples of a strange new class of infinite series he called "mock theta functions". It turns out all of these examples are essentially specializations of a so-called universal mock theta function $g_3(z,q)$ of Gordon-McIntosh. Here we show that $g_3$ arises naturally from the reciprocal of the classical Jacobi triple product -- and is intimately tied to rank generating functions for unimodal sequences, which are connected to mock modular and quantum modular forms -- under the action of an operator related to statistical physics and partition theory, the $q$-bracket of Bloch-Okounkov. Secondly, we find $g_3(z,q)$ to extend in $q$ to the entire complex plane minus the unit circle, and give a finite formula for this universal mock theta function at roots of unity, that is simple by comparison to other such formulas in the literature; we also indicate similar formulas for other $q$-hypergeometric series. Finally, we look at interesting "quantum" behaviors of mock theta functions inside, outside, and on the unit circle.

math.NT↗

arXiv2017

Phonon Sidebands in Transition Metal Dichalcogenides

Excitons dominate the optical properties of monolayer transition metal dichalcogenides (TMDs). Besides optically accessible bright exciton states, TMDs exhibit also a multitude of optically forbidden dark excitons. Here, we show that efficient exciton-phonon scattering couples bright and dark states and gives rise to an asymmetric excitonic line shape. The observed asymmetry can be traced back to phonon-induced sidebands that are accompanied by a polaron redshift. We present a joint theory-experiment study investigating the microscopic origin of these sidebands in different TMD materials taking into account intra- and intervalley scattering channels opened by optical and acoustic phonons. The gained insights contribute to a better understanding of the optical fingerprint of these technologically promising nanomaterials.

cond-mat.mes-hall↗

arXiv2017

Alternating "strange" functions

In this note we consider infinite series similar to the "strange" function $F(q)$ of Kontsevich studied by Zagier, Bryson-Ono-Pitman-Rhoades, Bringmann-Folsom-Rhoades, Rolen-Schneider, and others in connection to quantum modular forms. We show that a class of "strange" alternating series that are well-defined almost nowhere in the complex plane can be added (using a modified definition of limits) to familiar infinite products to produce convergent $q$-hypergeometric series, of a shape that specializes to Ramanujan's mock theta function $f(q)$, Zagier's quantum modular form $σ(q)$, and other interesting number-theoretic objects. We also discuss Cesàro sums for these alternating series, and continued fractions that are similarly "strange".

math.NT↗

arXiv2017

Partition-theoretic formulas for arithmetic densities

If $\gcd(r,t)=1$, then a theorem of Alladi offers the Möbius sum identity $$-\sum_{\substack{ n \geq 2 \\ p_{\rm{min}}(n) \equiv r \pmod{t}}} μ(n)n^{-1}= \frac{1}{φ(t)}. $$ Here $p_{\rm{min}}(n)$ is the smallest prime divisor of $n$. The right-hand side represents the proportion of primes in a fixed arithmetic progression modulo $t$. Locus generalized this to Chebotarev densities for Galois extensions. Answering a question of Alladi, we obtain analogs of these results to arithmetic densities of subsets of positive integers using $q$-series and integer partitions. For suitable subsets $§$ of the positive integers with density $d_§$, we prove that \[- \lim_{q \to 1} \sum_{\substack{ λ\in \mathcal{P} \\ \rm{sm}(λ) \in §}} μ_{\mathcal{P}} (λ)q^{\vert λ\vert} = d_§,\] where the sum is taken over integer partitions $λ$, $μ_{\mathcal{P}}(λ)$ is a partition-theoretic Möbius function, $\vert λ\vert$ is the size of partition $λ$, and $\rm{sm}(λ)$ is the smallest part of $λ$. In particular, we obtain partition-theoretic formulas for even powers of $π$ when considering power-free integers.

math.CO↗

arXiv2016

Fibonacci numbers and the golden ratio

In this expository paper written to commemorate Fibonacci Day 2016, we discuss famous relations involving the Fibonacci sequence, the golden ratio, continued fractions and nested radicals, and show how these fit into a more general framework stemming from the quadratic formula.

math.HO↗

arXiv2016

Arithmetic of partitions and the $q$-bracket operator

We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and find many theorems of classical number theory arise as particular cases of extremely general combinatorial structure laws. We then see that the relatively recently-defined $q$-bracket operator $\left _q$, studied by Bloch-Okounkov, Zagier, and others for its quasimodular properties, plays a deep role in the theory of partitions, quite apart from questions of modularity. Moreover, we give an explicit formula for the coefficients of $\left _q$ for any function $f$ defined on partitions, and, conversely, give a partition-theoretic function whose $q$-bracket is a given power series.

math.NT↗