We prove that the zero-divisor graph $\Gamma (P)$ of a Boolean poset P is both well-covered and Cohen–Macaulay. Furthermore, for a poset $\mathbf {P} = \prod _{i=1}^{n} P_i \ (n \ge 3)$, where each $P_i$ is a finite bounded poset satisfying $Z(P_i) = \{0\}$ for all i and $2 \le |P_1| \le |P_2| \le \cdots \le |P_n|,$ we show that the zero-divisor graph $\Gamma (\mathbf {P})$ is Cohen–Macaulay if and only if $\mathbf {P}$ is a Boolean lattice.

We study the congruence classes attained by positive integers D with a prescribed period for the continued fraction of $\sqrt D$. As an application, we refine the results on large ranks of universal quadratic forms over real quadratic fields by imposing congruence conditions on their discriminants.

Recently, Harrington et al. [‘Every arithmetic progression contains infinitely many b-Niven numbers’, Bull. Aust. Math. Soc. 109(3) (2024), 409–413] proved that every arithmetic progression contains infinitely many base-b Niven numbers for any fixed $b\ge 2$. We use a sparse repunit construction to treat a structured two-base version of the same problem, showing that every arithmetic progression with common difference relatively prime to b contains infinitely many integers that are simultaneously b-Niven and $b^k$-Niven (indeed, we can obtain simultaneous $b^\ell $-Niven-ness for $\ell =1,\ldots , k$).

Let $\triangle $ denote the integers represented by the quadratic form $x^2+3y^2$ and denote the numbers represented as a sum of two squares. For a nonzero integer a, let be the set of integers n such that $n \in \triangle $ and . We conduct a census of in short intervals by showing that there exists a constant $H_{a}> 0$ with

for large x. To derive this result and its generalisation, we use a theorem of Tolev [‘On the remainder term in the circle problem in an arithmetic progression’, Proc. Steklov Inst. Math. 276 (2012), 261–274].

We study a pair of finite multisets of nonnegative integers which have coinciding representation functions. We show that there is a close connection to factorisation of polynomials into cyclotomic polynomials. As an application, we give a complete characterisation of pairs of such sets whose union is the interval $[0,m]$ with two elements removed.

Poincaré series are of fundamental importance in the theory of modular forms. For a fixed even integer $k \ge 4$, the space of modular forms of weight k can be spanned by certain Poincaré series. We provide a criterion for a modular form to have a specified number of zeros on the unit circle in the standard fundamental domain for the action of $\mathrm {SL}(2,{\mathbb {Z}})$ on the complex upper half-plane. The criterion is given in terms of the coefficients that are obtained by writing the modular form as a linear combination of the generating Poincaré series.

Let $r\ge 1$ be an integer. For any multiple index $\mathbf {s}=(s_1,s_2,\ldots ,s_r) \in \mathbb {Z}_{\geq 1}^r$ with $s_r > 1$, the multiple zeta value (MZV for short) is defined by

$$ \begin{align*} \zeta(s_1,s_2,\ldots,s_r):=\sum_{1\leq k_1<k_2<\cdots<k_r} \frac{1}{k_1^{s_1}k_2^{s_2}\cdots k_r^{s_r}} \end{align*} $$

and the multiple t-value is defined by

$$ \begin{align*} t(s_1,s_2,\ldots,s_r):=\sum_{1\leq k_1<k_2<\cdots<k_r} \frac{1}{(2k_1-1)^{s_1}(2k_2-1)^{s_2}\cdots(2k_r-1)^{s_r}}. \end{align*} $$

If the index is empty, then we define the value $t(\emptyset ):=1$. We denote by $\{a_1,a_2,\ldots ,a_k\}^d$ the sequence formed by repeating the sequence $\{a_1,a_2,\ldots ,a_k\}$ exactly d times. Let $H(a,b)=\zeta (\{2\}^a,3,\{2\}^b)$ and $T(a,b):=t(\{2\}^a,3,\{2\}^b)$. By using the Lai–Lupu–Orr integral expressions for $H(a,b)$ and $T(a,b)$, and the properties of the Beta and Gamma functions, we show that for any nonnegative integers a and b,

$$ \begin{align*} H(a,b):=\frac{-4\pi^{2a+2b+2}}{(2a+2)!}\sum_{n=0}^{\infty} \frac{\zeta(2n)}{(2n+2a+2)(2n+2a+3)\cdots(2n+2a+2b+3)2^{2n}} \end{align*} $$

and

$$ \begin{align*} T(a,b)=\frac{-2}{(2a+1)!}\bigg(\frac{\pi}{2}\bigg)^{2a+2b+2} \sum_{n=0}^{\infty}\frac{\zeta(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}. \end{align*} $$

This confirms two conjectures proposed by Lupu [‘Another look at Zagier’s formula for multiple zeta values involving Hoffman elements’, Math. Z. 301 (2022), 3127–3140].

In this note, we study bounded functions of invariant average under multiplication. We show that if an arithmetic function is of moderate growth, then it is asymptotically uncorrelated to such functions. This provides several new disjointness results on the distribution of orbits along $\Omega (n)$.

Let $f(x) = (x^{2}+1)^{n} - ax^{n} \in \mathbb {Z}[x]$ and assume $f(x)$ is irreducible. Let $\theta $ be a root of $f(x)$, set $K= \mathbb {Q}(\theta )$ and denote by $\mathbb {Z}_{K}$ the ring of integers of K. The index of f, denoted $\operatorname {ind}(f)$, is the index of $\mathbb {Z}[\theta ]$ in $\mathbb {Z}_{K}$. A polynomial $f(x)$ is said to be monogenic if $\operatorname {ind}(f) = 1$. We compute the discriminant of the polynomial $f(x)$, and then derive necessary and sufficient conditions on the parameters a and n for $f(x)$ to be monogenic. Furthermore, we provide a complete description of the primes that divide $\operatorname {ind}(f)$.

We establish a criterion for the integrality of averages of roots of unity and apply it to settle a conjecture regarding the linearity of functions on $\mathbb {Z}_n$. Specifically, we prove that for any modulus $n \ge 1$, if $f: \mathbb {Z}_n \to \mathbb {Z}_n$ is a function whose averages $({1}{/n}) \sum _{x=0}^{n-1} \omega ^{f(x)+bx}$ (where $\omega =e^{2\pi i/n}$) are algebraic integers for all $b \in \mathbb {Z}_n$, then f is necessarily linear modulo n. This provides a short, elementary proof that works uniformly for all n and avoids the finite-field machinery used in previous partial results. Furthermore, when $n=p^r$, we use a local–global integrality argument to show that any normalised sum of $p^r$th roots of unity that is p-adically integral must be either $0$ or a single root of unity. As an application, we completely characterise the perfect isometries of the cyclic group $C_{p^r}$; they are precisely those induced by affine permutations $x \mapsto \alpha x + \beta $ with $\gcd (\alpha , p^r)=1$.

We explore the injectivity of the evaluation map $\mathrm {eva}_{f,\mathcal {A}} : \mathcal {A}^m \to \mathcal {A} $, where $ \mathcal {A} $ is an associative algebra over a field $ F $ and $ f $ is a polynomial in $ m \geq 1 $ variables with coefficients in F. Our investigation reveals that injectivity is possible only when $ m = 1 $ and $ f $ has degree one; for functions in two or more variables, such injectivity is impossible.

Let $F \ast G$ be a free product of a free group F and a LERF group G. We provide sufficient conditions for a subgroup H of $F \ast G$ to be $\mathcal A\cup \mathcal S$-separable, that is, for any finite set $\{\gamma _1, \ldots , \gamma _n\} \subset (F \ast G) \setminus H$, there is a surjection f from $F \ast G$ to an alternating or symmetric group such that $f(\gamma _i) \notin f(H)$ for all i. As a corollary, any finitely generated infinite-index subgroup of a free group is $\mathcal A\cup \mathcal S$-separable in the free product of the free group and an arbitrary LERF group, generalising a result of Wilton [‘Alternating quotients of free groups’, Enseign. Math. 58 (2012), 49–60].

Let $\mathcal {A}$ denote the class of normalised analytic functions f in the open unit disk $\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}$ with $f(0)=0$ and $f'(0)=1$. A function $f\in \mathcal {A}$ is said to be convex if $f(\mathbb {D})$ is convex. We establish a sharp upper bound for the third Hankel determinant corresponding to the inverse coefficients of convex univalent (that is, one-to-one) functions in the unit disk $\mathbb {D}$.

We give new analogues of Andrews’ celebrated spt-function identity using basic hypergeometric transformations due to Verma and Jain [‘Transformations between basic hypergeometric series on different bases and identities of Rogers–Ramanujan type’, J. Math. Anal. Appl. 76(1) (1980), 230–269].

We study the long-time behaviour of solutions to a free boundary model in river networks with small water flow speeds. First, we establish the well-posedness of our model. Under suitable assumptions, we then establish a spreading–vanishing dichotomy result for solutions to our model. By introducing a parameter $\sigma $ in the initial data, we derive sharp threshold behaviours.

We use a recent result of Orevkov and Pakovich [‘On intersection of lemniscates of rational functions’, Arnold J. Math. 11 (2025), 7–26] to obtain a bound on the frequency with which elements in an orbit of polynomial iterations can fall on the unit circle.