14 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
of integers . More precisely, let be a
conjugacy class of the Galois group of some finite Galois extension of
. Then we prove that
This theorem is a generalization of a result of Alladi from 1977 that asserts
that largest prime divisors are equidistributed in
arithmetic progressions modulo an integer , which occurs when is a
cyclotomic field
Congruences for powers of the partition function
Let denote the number of partitions of into colors. In
analogy with Ramanujan's work on the partition function, Lin recently proved in
\cite{Lin} that for every integer . Such
congruences, those of the form , were
previously studied by Kiming and Olsson. If is prime and , then such congruences satisfy . Inspired by Lin's example, we obtain natural infinite families of such
congruences. If (resp. and
) is prime and (resp.
and ), then for , where , 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 -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 is the phenomenon where an
infinite dimensional graded -module
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
. For each and each irreducible character
, we employ Frobenius' -character extension to define \textit{width McKay-Thompson
series} for ( copies) for each
-tuple in ( copies). These series are
modular functions which then reflect differences between -character values.
Furthermore, we establish orthogonality relations for the Frobenius
-characters, which dictate the compatibility of the extension of weak
moonshine for to width 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 . We
use this formula to prove recent conjectures of Chen concerning inequalities
which involve the partition function and . Further, we
strengthen one of the conjectures, and prove that for every there
is an effectively computable constant such that for all
, 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 , we must employ methods which are
completely different from those used by Lehmer to give effective error bounds
for . Instead, our approach relies on the fact that and
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
For a finite group , let denote the field generated over
by its character values. For , G. R. Robinson and J. G.
Thompson proved that where
. Confirming a speculation of Thompson, we show
that arbitrary suitable multiquadratic fields are similarly generated by the
values of -characters restricted to elements whose orders are only
divisible by ramified primes. To be more precise, we say that a -number is
a positive integer whose prime factors belong to a set of odd primes . Let be the field generated by the
values of -characters for even permutations whose orders are
-numbers. If , then we determine a constant with the
property that for all , we have
K_{\pi}(A_n)=\mathbb{Q}\left(\sqrt{p_1^*}, \sqrt{p_2^*},\dots,
\sqrt{p_t^*}\right).$
Generalized Paley graphs and their complete subgraphs of orders three and four
Let be an integer. Let be a prime power such that if is even, or, if is odd. The
generalized Paley graph of order , , is the graph with vertex set
where is an edge if and only if is a -th power
residue. We provide a formula, in terms of finite field hypergeometric
functions, for the number of complete subgraphs of order four contained in
, , which holds for all . This generalizes
the results of Evans, Pulham and Sheehan on the original (=2) Paley graph.
We also provide a formula, in terms of Jacobi sums, for the number of complete
subgraphs of order three contained in , . In
both cases we give explicit determinations of these formulae for small . We
show that zero values of (resp.
) yield lower bounds for the multicolor diagonal Ramsey
numbers (resp. ). We state explicitly these
lower bounds for small and compare to known bounds. We also examine the
relationship between both and ,
when is prime, and Fourier coefficients of modular forms
Fields generated by characters of finite linear groups
In previous work, the authors confirmed the speculation of J. G. Thompson
that certain multiquadratic fields are generated by specified character values
of sufficiently large alternating groups . Here we address the natural
generalization of this speculation to the finite general linear groups
and
.Comment: Minor revision, i.e. additional clarification in a few places, based
on the referee's repor
Non-standard binary representations and the Stern sequence
We show that the number of short binary signed-digit representations of an
integer is equal to the -th term in the Stern sequence. Various proofs
are provided, including direct, bijective, and generating function proofs. We
also show that this result can be derived from recent work of Monroe on binary
signed-digit representations of a fixed length
Representations of integers as quotients of sums of distinct powers of three
Which integers can be written as a quotient of sums of distinct powers of
three? We outline our first steps toward an answer to this question, beginning
with a necessary and almost sufficient condition. Then we discuss an algorithm
that indicates whether it is possible to represent a given integer as a
quotient of sums of distinct powers of three. When the given integer is
representable, this same algorithm generates all possible representations. We
develop a categorization of representations based on their connections to
-polynomials and give a complete description of the types of
representations for all integers up to 364. Finally, we discuss in detail the
representations of 7, 22, 34, 64, and 100, as well as some infinite families of
integers