If the determinant of a $3 \times 3$ matrix vanishes and its entries are unimodular complex numbers then two rows or two columns of the matrix are linearly dependent. The proof is remarkably easy. Generalizations include estimates for subdeterminants if the determinant is small and the moduli of the entries are close to $1$.

Let $(X,\rho,\mu)_{d,\theta}$ be a space of homogeneous type which includes metric measure spaces and some fractals, namely, $X$ is a set, $\rho$ is a quasi-metric on $X$ satisfying that there exist constants $C_0> 0$ and $\theta \in (0, 1]$ such that for all $x, x^{\prime}, y \in X$, $$|\rho(x, y) - \rho(x^{\prime}, y)| \leqslant C_{0}\rho(x, x^{\prime})^\theta[\rho(x, y) + \rho(x^{\prime}, y)]^{1-\theta},$$ and $\mu$ is a nonnegative Borel regular measure on $X$ satisfying that for some $d > 0$, all $x \in X$ and all $0 \lt r \lt \mathrm{diam} \ X$, $$\mu(\{y \in X \ : \ \rho(x,y) \lt r\}) \sim r^{d}.$$ In this paper, the authors establish the inhomogeneous discrete Calderón reproducing formulae on spaces of homogeneous type associated to a given special para-accretive function introduced by G. David, which will pave the way for developing the theory of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type associated to a given special para-accretive function.

Given an integer $\alpha \geqslant 2$, let $S_\alpha$ be the set of those positive integers $n$, with at least two distinct prime factors, which can be written as $n = \sum\limits_{p|n}p^\alpha$. We obtain general results concerning the nature of the sets $S_\alpha$ and we also identify all those $n \in S_3$ which have exactly three prime factors. We then consider the set $T \ (\mathrm{resp.} \ T_{0})$ of those positive integers $n$, with at least two distinct prime factors, which can be written as $n = \sum\limits_{p|n}p^{\alpha_{p}}$, where the exponents $\alpha_{p} \geqslant 1 \ (\mathrm{resp}. \ \alpha_{p} \geqslant 0)$ are allowed to vary with each prime factor $p$. We examine the size of $T(x) \ (\mathrm{resp}. \ T_{0}(x))$, the number of positive integers $n \leqslant x$ belonging to $T \ (\mathrm{resp}. \ T_{0})$.

In this paper we establish the boundedness in the context of Triebel-Lizorkin spaces for multilinear Marcinkiewicz operators.

Let $N_{g}(d)$ be the set of primes $p$ such that the order of $g$ modulo $p$, $\mathrm{ord}_p (g)$, is divisible by a prescribed integer $d$. Wiertelak showed that this set has a natural density, $\delta_{g}(d)$, with $\delta_{g}(d) \in \mathbb{Q}_{>0}$. Let $N_{g}(d)(x)$ be the number of primes $p \leqslant x$ that are in $N_{g}(d)$. A simple identity for $N_{g}(d)(x)$ is established. It is used to derive a more compact expression for $\delta_{g}(d)$ than known hitherto.

We give an improved upper bound for the error term in the Riemann-von Mangoldt formula for functions of degree 0 in the extended Selberg class. The bound is uniform for functions with no zeros in a certain half-plane. The first such bound was given by J. Kaczorowski and A. Pereili [3] in the proof that the extended Selberg class is a semigroup with factorization.

The aim of this paper is to study the diagonal embeddings of function spaces with dominating mixed smoothness. From certain point of view, this paper may be considered as a direct continuation of [8] and [6].

In the present paper we derive an asymptotic formula for $\sum\limits_{p \leqslant x,r_{k}(p)=q_{\tau}}1$, where $k$ is a product of different odd primes, $q_{\tau}$ is the $\tau$-th consecutive prime and $r_{k}(p)$ the least prime $q$ such that $(\mathrm{ord}_{p} \ q, k) = 1$.