Let $[t]$ be the integral part of the real number t and let $\mathbb {1}_{{\mathbb P}}$ be the characteristic function of the primes. Denote by $\pi _{\mathcal {S}}(x)$ the number of primes in the floor function set $\mathcal {S}(x) := \{[{x}/{n}] : 1\leqslant n\leqslant x\}$ and by $S_{\mathbb {1}_{{\mathbb P}}}(x)$ the number of primes in the sequence $\{[{x}/{n}]\}_{n\geqslant 1}$. Improving a result of Heyman [‘Primes in floor function sets’, Integers 22 (2022), Article no. A59], we show

$$ \begin{align*} \pi_{\mathcal{S}}(x) = \int_2^{\sqrt{x}} \frac{d t}{\log t} + \int_2^{\sqrt{x}} \frac{d t}{\log(x/t)} + O(\sqrt{x}\,\mathrm{e}^{-c(\log x)^{3/5}(\log\log x)^{-1/5}}) \quad\mbox{and}\quad S_{\mathbb{1}_{{\mathbb P}}}(x) = C_{\mathbb{1}_{{\mathbb P}}} x + O_{\varepsilon}(x^{9/19+\varepsilon}) \end{align*} $$

for $x\to \infty $, where $C_{\mathbb {1}_{{\mathbb P}}} := \sum _{p} {1}/{p(p+1)}$, $c>0$ is a positive constant and $\varepsilon $ is an arbitrarily small positive number.

Let D be a division ring and N be a subnormal subgroup of the multiplicative group $D^*$. We show that if N contains a nonabelian solvable subgroup, then N contains a nonabelian free subgroup.

In this paper, we show that every pair of sufficiently large even integers can be represented as a pair of eight prime cubes and k powers of $2$. In particular, we prove that $k=335$ is admissible, which improves the previous result.

For any Liouville number $\alpha $, all of the following are transcendental numbers: ${e}^\alpha $, $\log _{e}\alpha $, $\sin \alpha $, $\cos \alpha $, $\tan \alpha $, $\sinh \alpha $, $\cosh \alpha $, $\tanh \alpha $, $\arcsin \alpha $ and the inverse functions evaluated at $\alpha $ of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if ‘Liouville number’ is replaced by ‘U-number’, where U is one of Mahler’s classes of transcendental numbers.

We prove several finite product-sum identities involving the q-binomial coefficient, one of which is used to prove an amazing identity of Gauss. We then use this identity to evaluate certain quadratic Gauss sums and, together with known properties of quadratic Gauss sums, we prove the quadratic reciprocity law for the Jacobi symbol. We end our article with a new proof of Jenkins’ lemma, a lemma analogous to Gauss’ lemma. This article aims to show that Gauss’ amazing identity and the properties of quadratic Gauss sums are sufficient to establish the quadratic reciprocity law for the Jacobi symbol.

Suppose that R is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Using results from graded ring theory, we show that the associated Leavitt path algebra $L_R(E)$ is simple if and only if R is simple, $E^0$ has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra.

A partition of a positive integer n is called $\ell $-regular if none of its parts is divisible by $\ell $. Denote by $b_{\ell }(n)$ the number of $\ell $-regular partitions of n. We give a complete characterisation of the arithmetic of $b_{23}(n)$ modulo $11$ for all n not divisible by $11$ in terms of binary quadratic forms. Our result is obtained by establishing a relation between the generating function for these values of $b_{23}(n)$ and certain modular forms having complex multiplication by ${\mathbb Q}(\sqrt {-69})$.

We study the discriminants of the minimal polynomials $\mathcal {P}_n$ of the Ramanujan $t_n$ class invariants, which are defined for positive $n\equiv 11\pmod {24}$. We show that $\Delta (\mathcal {P}_n)$ divides $\Delta (H_n)$, where $H_n$ is the ring class polynomial, with quotient a perfect square and determine the sign of $\Delta (\mathcal {P}_n)$ based on the ideal class group structure of the order of discriminant $-n$. We also show that the discriminant of the number field generated by $j({(-1+\sqrt {-n})}/{2})$, where j is the j-invariant, divides $\Delta (\mathcal {P}_n)$. Moreover, using Ye’s computation of $\log|\Delta(H_n)|$ [‘Revisiting the Gross–Zagier discriminant formula’, Math. Nachr. 293 (2020), 1801–1826], we show that 3 never divides $\Delta(H_n)$, and thus $\Delta(\mathcal{P}_n)$, for all squarefree $n\equiv11\pmod{24}$.

We prove that if a solvable group A acts coprimely on a solvable group G, then A has a relatively ‘large’ orbit in its corresponding action on the set of ordinary complex irreducible characters of G. This improves an earlier result of Keller and Yang [‘Orbits of finite solvable groups on characters’, Israel J. Math. 199 (2014), 933–940].

We investigate uniform upper bounds for the number of powerful numbers in short intervals $(x, x + y]$. We obtain unconditional upper bounds $O({y}/{\log y})$ and $O(\kern1.3pt y^{11/12})$ for all powerful numbers and $y^{1/2}$-smooth powerful numbers, respectively. Conditional on the $abc$-conjecture, we prove the bound $O({y}/{\log ^{1+\epsilon } y})$ for squarefull numbers and the bound $O(\kern1.3pt y^{(2 + \epsilon )/k})$ for k-full numbers when $k \ge 3$. These bounds are related to Roth’s theorem on arithmetic progressions and the conjecture on the nonexistence of three consecutive squarefull numbers.

For a nonconstant elliptic surface over $\mathbb {P}^1$ defined over $\mathbb {Q}$, it is a result of Silverman [‘Heights and the specialization map for families of abelian varieties’, J. reine angew. Math. 342 (1983), 197–211] that the Mordell–Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is nonisotrivial, one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann [‘Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. I’, Math. Nachr. 49 (1971), 107–123] and Setzer [‘Elliptic curves of prime conductor’, J. Lond. Math. Soc. (2) 10 (1975), 367–378]. In this note, we show that the Legendre elliptic surface has the desired property, conditional on the existence of infinitely many Mersenne primes.

For a positive integer $r\geq 2$, a natural number n is r-free if there is no prime p such that $p^r\mid n$. Asymptotic formulae for the distribution of r-free integers in the floor function set $S(x):=\{\lfloor x/ n \rfloor :1\leq n\leq x\}$ are derived. The first formula uses an estimate for elements of $S(x)$ belonging to arithmetic progressions. The other, more refined, formula makes use of an exponent pair and the Riemann hypothesis.

Let G be a p-group for some prime p. Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G with order not equal to p. In this paper, we prove that if the Hughes subgroup of G is cyclic, then G has exponent p or is cyclic or is dihedral. We also prove that if the Hughes subgroup of G is generalised quaternion, then G must be generalised quaternion. With these results in hand, we classify the tidy p-groups.

A generating set S for a group G is independent if the subgroup generated by $S\setminus \{s\}$ is properly contained in G for all $s \in S.$ We describe the structure of finite groups G such that there are precisely two numbers appearing as the cardinalities of independent generating sets for G.

A partition $\lambda $ of n is said to be nearly self-conjugate if the Ferrers graph of $\lambda $ and its transpose have exactly $n-1$ cells in common. The generating function of the number of such partitions was first conjectured by Campbell and recently confirmed by Campbell and Chern (‘Nearly self-conjugate integer partitions’, submitted for publication). We present a simple and direct analytic proof and a combinatorial proof of an equivalent statement.

The central kernel $K(G)$ of a group G is the (characteristic) subgroup consisting of all elements $x\in G$ such that $x^{\gamma }=x$ for every central automorphism $\gamma $ of G. We prove that if G is a finite-by-nilpotent group whose central kernel has finite index, then the full automorphism group $Aut(G)$ of G is finite. Some applications of this result are given.

Gagola and Lewis [‘A character theoretic condition characterizing nilpotent groups’, Comm. Algebra 27 (1999), 1053–1056] proved that a finite group G is nilpotent if and only if $\chi (1)^{2}$ divides $\lvert G:\textrm {ker}\,\chi \rvert $ for every irreducible character $\chi $ of G. The theorem was later generalised by using monolithic characters. We generalise the theorem further considering only strongly monolithic characters. We also give some criteria for solvability and nilpotency of finite groups by their strongly monolithic characters.

We extend a result of Lieb [‘On the lowest eigenvalue of the Laplacian for the intersection of two domains’, Invent. Math. 74(3) (1983), 441–448] to the fractional setting. We prove that if A and B are two bounded domains in $\mathbb R^N$ and $\lambda _s(A)$, $\lambda _s(B)$ are the lowest eigenvalues of $(-\Delta )^s$, $0<s<1$, with Dirichlet boundary conditions, there exists some translation $B_x$ of B such that $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)$. Moreover, without the boundedness assumption on A and B, we show that for any $\varepsilon>0$, there exists some translation $B_x$ of B such that $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)+\varepsilon .$

We prove a reversed Hardy–Littlewood–Pólya inequality with finite terms. We also give the limit of the best constant.

The codegree of an irreducible character $\chi $ of a finite group G is $|G : \ker \chi |/\chi (1)$. We show that the Ree group ${}^2G_2(q)$, where $q = 3^{2f+1}$, is determined up to isomorphism by its set of codegrees.