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We extend spectral graph theory from the integral circulant graphs with prime power order to a Cayley graph over a finite chain ring and determine the spectrum and energy of such graphs. Moreover, we apply the results to obtain the energy of some gcd-graphs on a quotient ring of a unique factorisation domain.
Let
$P(n)$
denote the largest prime factor of an integer
$n\geq 2$
. In this paper, we study the distribution of the sequence
$\{f(P(n)):n\geq 1\}$
over the set of congruence classes modulo an integer
$b\geq 2$
, where
$f$
is a strongly
$q$
-additive integer-valued function (that is,
$f(aq^{j}+b)=f(a)+f(b),$
with
$(a,b,j)\in \mathbb{N}^{3}$
,
$0\leq b<q^{j}$
). We also show that the sequence
$\{{\it\alpha}P(n):n\geq 1,f(P(n))\equiv a\;(\text{mod}~b)\}$
is uniformly distributed modulo 1 if and only if
${\it\alpha}\in \mathbb{R}\!\setminus \!\mathbb{Q}$
.
We prove the unimodality of some coloured
$q$
-Eulerian polynomials, which involve the flag excedances, the major index and the fixed points on coloured permutation groups, via two recurrence formulas. In particular, we confirm a recent conjecture of Mongelli about the unimodality of the flag excedances over type B derangements. Furthermore, we find the coloured version of Gessel’s hook factorisation, which enables us to interpret these two recurrences combinatorially. We also provide a combinatorial proof of a symmetric and unimodal expansion for the coloured derangement polynomial, which was first established by Shin and Zeng using continued fractions.
We give three identities involving multiple zeta values of height one and of maximal height: an explicit formula for the height-one multiple zeta values, a regularised sum formula and a sum formula for the multiple zeta values of maximal height.
modulo a prime
$p$
, with variables
$1\leq x_{i}\leq h$
,
$i=1,\ldots ,{\it\nu}$
and arbitrary integers
$s_{j},{\it\lambda}_{j}$
,
$j=1,\ldots ,m$
, for a parameter
$h$
significantly smaller than
$p$
. We also mention some applications of this bound.
Let
$G$
be a finite group and
${\rm\Gamma}$
a
$G$
-symmetric graph. Suppose that
$G$
is imprimitive on
$V({\rm\Gamma})$
with
$B$
a block of imprimitivity and
${\mathcal{B}}:=\{B^{g};g\in G\}$
a system of imprimitivity of
$G$
on
$V({\rm\Gamma})$
. Define
${\rm\Gamma}_{{\mathcal{B}}}$
to be the graph with vertex set
${\mathcal{B}}$
such that two blocks
$B,C\in {\mathcal{B}}$
are adjacent if and only if there exists at least one edge of
${\rm\Gamma}$
joining a vertex in
$B$
and a vertex in
$C$
. Xu and Zhou [‘Symmetric graphs with 2-arc-transitive quotients’, J. Aust. Math. Soc.96 (2014), 275–288] obtained necessary conditions under which the graph
${\rm\Gamma}_{{\mathcal{B}}}$
is 2-arc-transitive. In this paper, we completely settle one of the cases defined by certain parameters connected to
${\rm\Gamma}$
and
${\mathcal{B}}$
and show that there is a unique graph corresponding to this case.
In this note, we prove that for any
${\it\nu}>0$
, there is no lacunary entire function
$f(z)\in \mathbb{Q}[[z]]$
such that
$f(\mathbb{Q})\subseteq \mathbb{Q}$
and
$\text{den}f(p/q)\ll q^{{\it\nu}}$
, for all sufficiently large
$q$
.
We prove an asymptotic formula for the sum
$\sum _{n\leq N}d(n^{2}-1)$
, where
$d(n)$
denotes the number of divisors of
$n$
. During the course of our proof, we also furnish an asymptotic formula for the sum
$\sum _{d\leq N}g(d)$
, where
$g(d)$
denotes the number of solutions
$x$
in
$\mathbb{Z}_{d}$
to the equation
$x^{2}\equiv 1~(\text{mod}~d)$
.
In this paper, by using the theory of reproducing kernel Hilbert spaces and the pair correlation formula constructed by Chandee et al. [‘Simple zeros of primitive Dirichlet
$L$
-functions and the asymptotic large sieve’, Q. J. Math.65(1) (2014), 63–87], we prove that at least 93.22% of low-lying zeros of primitive Dirichlet
$L$
-functions are simple in a proper sense, under the assumption of the generalised Riemann hypothesis.
We examine the tail distributions of integer partition ranks and cranks by investigating tail moments, which are analogous to the positive moments introduced by Andrews et al. [‘The odd moments of ranks and cranks’, J. Combin. Theory Ser. A120(1) (2013), 77–91].
Let
$K$
be a number field with ring of integers
${\mathcal{O}}$
. After introducing a suitable notion of density for subsets of
${\mathcal{O}}$
, generalising the natural density for subsets of
$\mathbb{Z}$
, we show that the density of the set of coprime
$m$
-tuples of algebraic integers is
$1/{\it\zeta}_{K}(m)$
, where
${\it\zeta}_{K}$
is the Dedekind zeta function of
$K$
. This generalises a result found independently by Mertens [‘Ueber einige asymptotische Gesetze der Zahlentheorie’, J. reine angew. Math.77 (1874), 289–338] and Cesàro [‘Question 75 (solution)’, Mathesis3 (1883), 224–225] concerning the density of coprime pairs of integers in
$\mathbb{Z}$
.
We study transcendence properties of certain infinite products of cyclotomic polynomials. In particular, we determine all cases in which the product is hypertranscendental. We then use various results from Mahler’s transcendence method to obtain algebraic independence results on such functions and their values.
We generalise a result of Hilbert which asserts that the Riemann zeta-function
${\it\zeta}(s)$
is hypertranscendental over
$\mathbb{C}(s)$
. Let
${\it\pi}$
be any irreducible cuspidal automorphic representation of
$\text{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$
with unitary central character. We establish a certain type of functional difference–differential independence for the associated
$L$
-function
$L(s,{\it\pi})$
. This result implies algebraic difference–differential independence of
$L(s,{\it\pi})$
over
$\mathbb{C}(s)$
(and more strongly, over a certain field
${\mathcal{F}}_{s}$
which contains
$\mathbb{C}(s)$
). In particular,
$L(s,{\it\pi})$
is hypertranscendental over
$\mathbb{C}(s)$
. We also extend a result of Ostrowski on the hypertranscendence of ordinary Dirichlet series.
Let
$m$
be a positive integer and
$p$
a prime number. We prove the orthogonality of some character sums over the finite field
$\mathbb{F}_{p^{m}}$
or over a subset of a finite field and use this to construct some new approximately mutually unbiased bases of dimension
$p^{m}$
over the complex number field
$\mathbb{C}$
, especially with
$p=2$
.
The subgroup commutativity degree of a group
$G$
is the probability that two subgroups of
$G$
commute, or equivalently that the product of two subgroups is again a subgroup. For the dihedral, quasi-dihedral and generalised quaternion groups (all of 2-power cardinality), the subgroup commutativity degree tends to 0 as the size of the group tends to infinity. This also holds for the family of projective special linear groups over fields of even characteristic and for the family of the simple Suzuki groups. In this short note, we show that the family of finite
$P$
-groups also has this property.
Using the inductive structure of a Fermat variety by Shioda and Katsura [‘On Fermat varieties’, Tohoku Math. J. (2) 31(1) (1979), 97–115], we estimate the refined motivic dimension of certain Fermat varieties. As an application of our computation, we present an elementary proof of the generalised Hodge conjecture for those varieties.
Let
$b_{3,5}(n)$
denote the number of partitions of
$n$
into parts that are not multiples of 3 or 5. We establish several infinite families of congruences modulo 2 for
$b_{3,5}(n)$
. In the process, we also prove numerous parity results for broken 7-diamond partitions.
A subset
$X$
of a group
$G$
is a set of pairwise noncommuting elements if
$ab\neq ba$
for any two distinct elements
$a$
and
$b$
in
$X$
. If
$|X|\geq |Y|$
for any other set of pairwise noncommuting elements
$Y$
in
$G$
, then
$X$
is called a maximal subset of pairwise noncommuting elements and the cardinality of such a subset (if it exists) is denoted by
${\it\omega}(G)$
. In this paper, among other things, we prove that, for each positive integer
$n$
, there are only finitely many groups
$G$
, up to isoclinism, with
${\it\omega}(G)=n$
, and we obtain similar results for groups with exactly
$n$
centralisers.
In this paper we prove the following result: let
$m,n\geq 1$
be distinct integers, let
$R$
be an
$mn(m+n)|m-n|$
-torsion free semiprime ring and let
$D:R\rightarrow R$
be an
$(m,n)$
-Jordan derivation, that is an additive mapping satisfying the relation
$(m+n)D(x^{2})=2mD(x)x+2nxD(x)$
for
$x\in R$
. Then
$D$
is a derivation which maps
$R$
into its centre.
Let
$b_{\ell }(n)$
denote the number of
$\ell$
-regular partitions of
$n$
. In this paper we establish a formula for
$b_{13}(3n+1)$
modulo
$3$
and use this to find exact criteria for the
$3$
-divisibility of
$b_{13}(3n+1)$
and
$b_{13}(3n)$
. We also give analogous criteria for
$b_{7}(3n)$
and
$b_{7}(3n+2)$
.