The $\lambda $-quiddities of size n are n-tuples of elements of a fixed set, solutions of a matrix equation appearing in the study of Coxeter’s friezes. Their number and properties are closely linked to the structure and the cardinality of the chosen set. Our main objective is an explicit formula giving the number of $\lambda $-quiddities of odd size, and a lower and upper bound for the number of $\lambda $-quiddities of even size, over the rings ${\mathbb {Z}}/2^{m}{\mathbb {Z}}$ ($m \geq 2$). We also give explicit formulae for the number of $\lambda $-quiddities of size n over ${\mathbb {Z}}/8{\mathbb {Z}}$.

For any positive integer n, let $\sigma (n)$ be the sum of all positive divisors of n. We prove that for every integer k with $1\leq k\leq 29$ and $(k,30)=1,$

$$ \begin{align*} \sum_{n\leq K}\sigma(30n)>\sum_{n\leq K}\sigma(30n+k) \end{align*} $$

for all $K\in \mathbb {N},$ which gives a positive answer to a problem posed by Pongsriiam [‘Sums of divisors on arithmetic progressions’, Period. Math. Hungar. 88 (2024), 443–460].

Ishitsuka et al. [‘Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic’, Int. J. Number Theory 16(4) (2020), 881–905] found all points on the Fermat quartic ${F_4\colon x^4+y^4=z^4}$ over quadratic extensions of ${\mathbb {Q}}(\zeta _8)$, where $\zeta _8$ is the eighth primitive root of unity $e^{i\pi /4}$. Using Mordell’s technique, we give an alternative proof for the result of Ishitsuka et al. and extend it to the rational function field ${\mathbb {Q}}({\zeta _8})(T_1,T_2,\ldots ,T_n)$.

We investigate the discrepancy between the distributions of the random variable $\log L (\sigma , f \times f, X)$ and that of $\log L(\sigma +it, f \times f)$, that is,

$$ \begin{align*} D_{\sigma} (T) := \sup_{\mathcal{R}} |\mathbb{P}_T(\log L(\sigma+it, f \times f) \in \mathcal{R}) - \mathbb{P}(\log L(\sigma, f \times f, X) \in \mathcal{R})|, \end{align*} $$

where the supremum is taken over rectangles $\mathcal {R}$ with sides parallel to the coordinate axes. For fixed $T>3$ and $2/3 <\sigma _0 < \sigma < 1$, we prove that

$$ \begin{align*} D_{\sigma} (T) \ll \frac{1}{(\log T)^{\sigma}}. \end{align*} $$

We establish upper bounds for moments of smoothed quadratic Dirichlet character sums under the generalized Riemann hypothesis, confirming a conjecture of M. Jutila [‘On sums of real characters’, Tr. Mat. Inst. Steklova 132 (1973), 247–250].

Let $\mathcal {P}$ be the set of primes and $\pi (x)$ the number of primes not exceeding x. Let $P^+(n)$ be the largest prime factor of n, with the convention $P^+(1)=1$, and $ T_c(x)=\#\{p\le x:p\in \mathcal {P},P^+(p-1)\ge p^c\}. $ Motivated by a conjecture of Chen and Chen [‘On the largest prime factor of shifted primes’, Acta Math. Sin. (Engl. Ser.) 33 (2017), 377–382], we show that for any c with $8/9\le c<1$,

$$ \begin{align*} \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\le 8(1/c-1), \end{align*} $$

which clearly means that

$$ \begin{align*} \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\rightarrow 0 \quad \text{as } c\rightarrow 1. \end{align*} $$

We establish some inequalities that arise from truncating Lerch sums and derive uniform asymptotic formulae for the spt-crank of ordinary partitions. The uniform asymptotic formulae improve upon a result of Mao [‘Asymptotic formulas for spt-crank of partitions’, J. Math. Anal. Appl. 460(1) (2018), 121–139].

An integer partition of a positive integer n is called t-core if none of its hook lengths is divisible by t. Gireesh et al. [‘A new analogue of t-core partitions’, Acta Arith. 199 (2021), 33–53] introduced an analogue $\overline {a}_t(n)$ of the t-core partition function. They obtained multiplicative formulae and arithmetic identities for $\overline {a}_t(n)$ where $t \in \{3,4,5,8\}$ and studied the arithmetic density of $\overline {a}_t(n)$ modulo $p_i^{\,j}$ where $t=p_1^{a_1}\cdots p_m^{a_m}$ and $p_i\geq 5$ are primes. Bandyopadhyay and Baruah [‘Arithmetic identities for some analogs of the 5-core partition function’, J. Integer Seq. 27 (2024), Article no. 24.4.5] proved new arithmetic identities satisfied by $\overline {a}_5(n)$. We study the arithmetic densities of $\overline {a}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^\alpha m$ where $\gcd (m,6)$=1. Also, employing a result of Ono and Taguchi [‘2-adic properties of certain modular forms and their applications to arithmetic functions’, Int. J. Number Theory 1 (2005), 75–101] on the nilpotency of Hecke operators, we prove an infinite family of congruences for $\overline {a}_3(n)$ modulo arbitrary powers of 2.

Let M be a Puiseux monoid, that is, a monoid consisting of nonnegative rationals (under standard addition). In this paper, we study factorisations in atomic Puiseux monoids through the lens of their associated Betti graphs. The Betti graph of $b \in M$ is the graph whose vertices are the factorisations of b with edges between factorisations that share at least one atom. If the Betti graph associated to b is disconnected, then we call b a Betti element of M. We explicitly compute the set of Betti elements for a large class of Puiseux monoids (the atomisations of certain infinite sequences of rationals). The process of atomisation is quite useful in studying the arithmetic of Puiseux monoids, and it has been actively considered in recent literature. This leads to an argument that for every positive integer n, there exists an atomic Puiseux monoid with exactly n Betti elements.

We present a new version of a generalisation to elliptic nets of a theorem of Ward [‘Memoir on elliptic divisibility sequences’, Amer. J. Math. 70 (1948), 31–74] on symmetry of elliptic divisibility sequences. Our results cover all that is known today.

We find an upper bound for the number of groups of order n up to isomorphism in the variety ${\mathfrak {S}}={\mathfrak {A}_p}{\mathfrak {A}_q}{\mathfrak {A}_r}$, where p, q and r are distinct primes. We also find a bound on the orders and on the number of conjugacy classes of subgroups that are maximal amongst the subgroups of the general linear group that are also in the variety $\mathfrak {A}_q\mathfrak {A}_r$.

We prove that centralisers of elements in [finitely generated free]-by-cyclic groups are computable. As a corollary, given two conjugate elements in a [finitely generated free]-by-cyclic group, the set of conjugators can be computed and the conjugacy problem with context-free constraints is decidable. We pose several problems arising naturally from this work.

We obtain sharp bounds for the second Hankel determinant of logarithmic inverse coefficients for starlike and convex functions.

We study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols $\varphi $ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.

Liu [‘Supercongruences for truncated Appell series’, Colloq. Math. 158(2) (2019), 255–263] and Lin and Liu [‘Congruences for the truncated Appell series $F_3$ and $F_4$’, Integral Transforms Spec. Funct. 31(1) (2020), 10–17] confirmed four supercongruences for truncated Appell series. Motivated by their work, we give a new supercongruence for the truncated Appell series $F_{1}$, together with two generalisations of this supercongruence, by establishing its q-analogues.

We provide well-posedness results for nonlinear parabolic partial differential equations (PDEs) given by reaction–diffusion equations describing the concentration of oxygen in encapsulated cells. The cells are described in terms of a core and a shell, which introduces a discontinuous diffusion coefficient as the material properties of the core and shell differ. In addition, the cells are subject to general nonlinear consumption of oxygen. As no monotonicity condition is imposed on the consumption, monotone operator theory cannot be used. Moreover, the discontinuity in the diffusion coefficient bars us from applying classical results on strong solutions. However, by directly applying a Galerkin method, we obtain uniqueness and existence of the strong form solution. These results provide the basis to study the dynamics of cells in critical states.

We give a sharp estimate for the first eigenvalue of the Schrödinger operator $L:=-\Delta -\sigma $ which is defined on the closed minimal submanifold $M^{n}$ in the unit sphere $\mathbb {S}^{n+m}$, where $\sigma $ is the square norm of the second fundamental form.