931,437 research outputs found

### A new formula for Chebotarev densities

We give a new formula for the Chebotarev densities of Frobenius elements in
Galois groups. This formula is given in terms of smallest prime factors
$p_{\mathrm{min}}(n)$ of integers $n\geq2$. More precisely, let $C$ be a
conjugacy class of the Galois group of some finite Galois extension $K$ of
$\mathbb{Q}$. Then we prove that
$-\lim_{X\rightarrow\infty}\sum_{\substack{2\leq n\leq
X\\[1pt]\left[\frac{K/\mathbb{Q}}{p_{\mathrm{min}}(n)}\right]=C}}\frac{\mu(n)}{n}=\frac{\#C}{\#G}.$
This theorem is a generalization of a result of Alladi from 1977 that asserts
that largest prime divisors $p_{\mathrm{max}}(n)$ are equidistributed in
arithmetic progressions modulo an integer $k$, which occurs when $K$ is a
cyclotomic field $\mathbb{Q}(\zeta_k)$

### Congruences for powers of the partition function

Let $p_{-t}(n)$ denote the number of partitions of $n$ into $t$ colors. In
analogy with Ramanujan's work on the partition function, Lin recently proved in
\cite{Lin} that $p_{-3}(11n+7)\equiv0\pmod{11}$ for every integer $n$. Such
congruences, those of the form $p_{-t}(\ell n + a) \equiv 0 \pmod {\ell}$, were
previously studied by Kiming and Olsson. If $\ell \geq 5$ is prime and $-t \not
\in \{\ell - 1, \ell -3\}$, then such congruences satisfy $24a \equiv -t \pmod
{\ell}$. Inspired by Lin's example, we obtain natural infinite families of such
congruences. If $\ell\equiv2\pmod{3}$ (resp. $\ell\equiv3\pmod{4}$ and
$\ell\equiv11\pmod{12}$) is prime and $r\in\{4,8,14\}$ (resp. $r\in\{6,10\}$
and $r=26$), then for $t=\ell s-r$, where $s\geq0$, we have that
\begin{equation*} p_{-t}\left(\ell
n+\frac{r(\ell^2-1)}{24}-\ell\Big\lfloor\frac{r(\ell^2-1)}{24\ell}\Big\rfloor\right)\equiv0\pmod{\ell}.
\end{equation*} Moreover, we exhibit infinite families where such congruences
cannot hold

### Combinatorial Properties of Rogers-Ramanujan-Type Identities Arising from Hall-Littlewood Polynomials

Here we consider the $q$-series coming from the Hall-Littlewood polynomials,
\begin{equation*} R_\nu(a,b;q)=\sum_{\substack{\lambda \\[1pt] \lambda_1\leq
a}} q^{c|\lambda|} P_{2\lambda}\big(1,q,q^2,\dots;q^{2b+d}\big).
\end{equation*} These series were defined by Griffin, Ono, and Warnaar in their
work on the framework of the Rogers-Ramanujan identities. We devise a recursive
method for computing the coefficients of these series when they arise within
the Rogers-Ramanujan framework. Furthermore, we study the congruence properties
of certain quotients and products of these series, generalizing the famous
Ramanujan congruence \begin{equation*} p(5n+4)\equiv0\pmod{5}. \end{equation*}Comment: 16 pages v2: Minor changes included, to appear in Annals of
Combinatoric

### Higher Width Moonshine

\textit{Weak moonshine} for a finite group $G$ is the phenomenon where an
infinite dimensional graded $G$-module $V_G=\bigoplus_{n\gg-\infty}V_G(n)$
has the property that its trace functions, known as McKay-Thompson series, are
modular functions. Recent work by DeHority, Gonzalez, Vafa, and Van Peski
established that weak moonshine holds for every finite group. Since weak
moonshine only relies on character tables, which are not isomorphism class
invariants, non-isomorphic groups can have the same McKay-Thompson series. We
address this problem by extending weak moonshine to arbitrary width
$s\in\mathbb{Z}^+$. For each $1\leq r\leq s$ and each irreducible character
$\chi_i$, we employ Frobenius' $r$-character extension $\chi_i^{(r)} \colon
G^{(r)}\rightarrow\mathbb{C}$ to define \textit{width $r$ McKay-Thompson
series} for $V_G^{(r)}:=V_G\times\cdots\times V_G$ ($r$ copies) for each
$r$-tuple in $G^{(r)}:=G\times\cdots\times G$ ($r$ copies). These series are
modular functions which then reflect differences between $r$-character values.
Furthermore, we establish orthogonality relations for the Frobenius
$r$-characters, which dictate the compatibility of the extension of weak
moonshine for $V_G$ to width $s$ weak moonshine.Comment: Versions 2 and 3 address comments from the referee

### Effective Bounds for the Andrews spt-function

In this paper, we establish an asymptotic formula with an effective bound on
the error term for the Andrews smallest parts function $\mathrm{spt}(n)$. We
use this formula to prove recent conjectures of Chen concerning inequalities
which involve the partition function $p(n)$ and $\mathrm{spt}(n)$. Further, we
strengthen one of the conjectures, and prove that for every $\epsilon>0$ there
is an effectively computable constant $N(\epsilon) > 0$ such that for all
$n\geq N(\epsilon)$, we have \begin{equation*}
\frac{\sqrt{6}}{\pi}\sqrt{n}\,p(n)<\mathrm{spt}(n)<\left(\frac{\sqrt{6}}{\pi}+\epsilon\right)
\sqrt{n}\,p(n). \end{equation*} Due to the conditional convergence of the
Rademacher-type formula for $\mathrm{spt}(n)$, we must employ methods which are
completely different from those used by Lehmer to give effective error bounds
for $p(n)$. Instead, our approach relies on the fact that $p(n)$ and
$\mathrm{spt}(n)$ can be expressed as traces of singular moduli.Comment: Changed the title. Added more details and simplified some arguments
in Section

### Multiquadratic fields generated by characters of $A_n$

For a finite group $G$, let $K(G)$ denote the field generated over
$\mathbb{Q}$ by its character values. For $n>24$, G. R. Robinson and J. G.
Thompson proved that $K(A_n)=\mathbb{Q}\left (\{ \sqrt{p^*} \ : \ p\leq n \
{\text{ an odd prime with } p\neq n-2}\}\right),$ where
$p^*:=(-1)^{\frac{p-1}{2}}p$. Confirming a speculation of Thompson, we show
that arbitrary suitable multiquadratic fields are similarly generated by the
values of $A_n$-characters restricted to elements whose orders are only
divisible by ramified primes. To be more precise, we say that a $\pi$-number is
a positive integer whose prime factors belong to a set of odd primes $\pi:=
\{p_1, p_2,\dots, p_t\}$. Let $K_{\pi}(A_n)$ be the field generated by the
values of $A_n$-characters for even permutations whose orders are
$\pi$-numbers. If $t\geq 2$, then we determine a constant $N_{\pi}$ with the
property that for all $n> N_{\pi}$, we have
$$K_{\pi}(A_n)=\mathbb{Q}\left(\sqrt{p_1^*}, \sqrt{p_2^*},\dots,
\sqrt{p_t^*}\right).$

### Speaking Rate Effects on Locus Equation Slope

A locus equation describes a 1st order regression fit to a scatter of vowel steady-state frequency values predicting vowel onset frequency values. Locus equation coefficients are often interpreted as indices of coarticulation. Speaking rate variations with a constant consonant–vowel form are thought to induce changes in the degree of coarticulation. In the current work, the hypothesis that locus slope is a transparent index of coarticulation is examined through the analysis of acoustic samples of large-scale, nearly continuous variations in speaking rate. Following the methodological conventions for locus equation derivation, data pooled across ten vowels yield locus equation slopes that are mostly consistent with the hypothesis that locus equations vary systematically with coarticulation. Comparable analyses between different four-vowel pools reveal variations in the locus slope range and changes in locus slope sensitivity to rate change. Analyses across rate but within vowels are substantially less consistent with the locus hypothesis. Taken together, these findings suggest that the practice of vowel pooling exerts a non-negligible influence on locus outcomes. Results are discussed within the context of articulatory accounts of locus equations and the effects of speaking rate change

### Confirmatory factor analysis and invariance testing between Blacks and Whites of the Multidimensional Health Locus of Control scale.

The factor structure of the Multidimensional Health Locus of Control scale remains in question. Additionally, research on health belief differences between Black and White respondents suggests that the Multidimensional Health Locus of Control scale may not be invariant. We reviewed the literature regarding the latent variable structure of the Multidimensional Health Locus of Control scale, used confirmatory factor analysis to confirm the three-factor structure of the Multidimensional Health Locus of Control, and analyzed between-group differences in the Multidimensional Health Locus of Control structure and means across Black and White respondents. Our results indicate differences in means and structure, indicating more research is needed to inform decisions regarding whether and how to deploy the Multidimensional Health Locus of Control appropriately

### On isotropic divisors on irreducible symplectic manifolds

Let X be an irreducible symplectic manifold and L a divisor on X. Assume that
L is isotropic with respect to the Beauville-Bogomolov quadratic form. We
define the rational Lagrangian locus and the movable locus on the universal
deformation space of the pair (X, L). We prove that the rational Lagrangian
locus is empty or coincide with the movable locus

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