Using a recent breakthrough of Smith [18], we improve the results of Fouvry and Klüners [4, 5] on the solubility of the negative Pell equation. Let $\mathcal {D}$ denote the set of positive squarefree integers having no prime factors congruent to $3$ modulo $4$. Stevenhagen [19] conjectured that the density of d in $\mathcal {D}$ such that the negative Pell equation $x^2-dy^2=-1$ is solvable with $x, y \in \mathbb {Z}$ is $58.1\%$, to the nearest tenth of a percent. By studying the distribution of the $8$-rank of narrow class groups $\operatorname {\mathrm {Cl}}^+(d)$ of $\mathbb {Q}(\sqrt {d})$, we prove that the infimum of this density is at least $53.8\%$.

We prove that analogues of the Hardy–Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula for the number of integers $n=p_1p_2 \leq X$ such that $n+h$ is a product of exactly two primes which holds for almost all $|h|\leq H$ with $\log^{19+\varepsilon}X\leq H\leq X^{1-\varepsilon}$, under a restriction on the size of one of the prime factors of n and $n+h$. Additionally, we consider correlations $n,n+h$ where n is a prime and $n+h$ has exactly two prime factors, establishing an asymptotic formula which holds for almost all $|h| \leq H$ with $X^{1/6+\varepsilon}\leq H\leq X^{1-\varepsilon}$.

The Euler–Mascheroni constant $\gamma =0.5772\ldots \!$ is the $K={\mathbb Q}$ example of an Euler–Kronecker constant $\gamma _K$ of a number field $K.$ In this note, we consider the size of the $\gamma _q=\gamma _{K_q}$ for cyclotomic fields $K_q:={\mathbb Q}(\zeta _q).$ Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that

$$ \begin{align*} \frac{1}{Q}\sum_{Q<q\le 2Q} |{\gamma_q - \log q}| = o(\log Q).\end{align*} $$

In other words, under EH, the $\gamma _q /\!\log q$ in these ranges converge to the one point distribution at $1$. This theorem refines and extends a previous result of Ford, Luca and Moree for prime $q.$ The proof of this result is a straightforward modification of earlier work of Fouvry under the assumption of EH.

We compute, via motivic wall-crossing, the generating function of virtual motives of the Quot scheme of points on ${\mathbb{A}}^3$, generalising to higher rank a result of Behrend–Bryan–Szendrői. We show that this motivic partition function converges to a Gaussian distribution, extending a result of Morrison.

We prove that the Riemann hypothesis is equivalent to the condition $\int _{2}^x\left (\pi (t)-\operatorname {\textrm {li}}(t)\right )\textrm {d}t<0$ for all $x>2$. Here, $\pi (t)$ is the prime-counting function and $\operatorname {\textrm {li}}(t)$ is the logarithmic integral. This makes explicit a claim of Pintz. Moreover, we prove an analogous result for the Chebyshev function $\theta (t)$ and discuss the extent to which one can make related claims unconditionally.

Let $g \geq 1$ be an integer and let $A/\mathbb Q$ be an abelian variety that is isogenous over $\mathbb Q$ to a product of g elliptic curves defined over $\mathbb Q$, pairwise non-isogenous over $\overline {\mathbb Q}$ and each without complex multiplication. For an integer t and a positive real number x, denote by $\pi _A(x, t)$ the number of primes $p \leq x$, of good reduction for A, for which the Frobenius trace $a_{1, p}(A)$ associated to the reduction of A modulo p equals t. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that $\pi _A(x, 0) \ll _A x^{1 - \frac {1}{3 g+1 }}/(\operatorname {log} x)^{1 - \frac {2}{3 g+1}}$ and $\pi _A(x, t) \ll _A x^{1 - \frac {1}{3 g + 2}}/(\operatorname {log} x)^{1 - \frac {2}{3 g + 2}}$ if $t \neq 0$. These bounds largely improve upon recent ones obtained for $g = 2$ by Chen, Jones, and Serban, and may be viewed as generalizations to arbitrary g of the bounds obtained for $g=1$ by Murty, Murty, and Saradha, combined with a refinement in the power of $\operatorname {log} x$ by Zywina. Under the assumptions stated above, we also prove the existence of a density one set of primes p satisfying $|a_{1, p}(A)|>p^{\frac {1}{3 g + 1} - \varepsilon }$ for any fixed $\varepsilon>0$.

Let $[t]$ be the integral part of the real number t. We study the distribution of the elements of the set $\mathcal {S}(x) := \{[{x}/{n}] : 1\leqslant n\leqslant x\}$ in the arithmetical progression $\{a+dq\}_{d\geqslant 0}$. We give an asymptotic formula

$$ \begin{align*} S(x; q, a) := \sum_{\substack{m\in \mathcal{S}(x)\\ m\equiv a \pmod q}} 1 = \frac{2\sqrt{x}}{q} + O((x/q)^{1/3}\log x), \end{align*} $$

which holds uniformly for $x\geqslant 3$, $1\leqslant q\leqslant x^{1/4}/(\log x)^{3/2}$ and $1\leqslant a\leqslant q$, where the implied constant is absolute. The special case $S(x; q, q)$ confirms a recent numerical test of Heyman [‘Cardinality of a floor function set’, Integers 19 (2019), Article no. A67].

We prove uniform bounds for the Petersson norm of the cuspidal part of the theta series. This gives an improved asymptotic formula for the number of representations by a quadratic form. As an application, we show that every integer $n \neq 0,4,7 \,(\textrm{mod}\ 8)$ is represented as $n= x_1^2 + x_2^2 + x_3^3$ for integers $x_1,x_2,x_3$ such that the product $x_1x_2x_3$ has at most 72 prime divisors.

In 1946, Erdős and Niven proved that no two partial sums of the harmonic series can be equal. We present a generalisation of the Erdős–Niven theorem by showing that no two partial sums of the series $\sum _{k=0}^\infty {1}/{(a+bk)}$ can be equal, where a and b are positive integers. The proof of our result uses analytic and p-adic methods.

We investigate, for given positive integers a and b, the least positive integer $c=c(a,b)$ such that the quotient $\varphi (c!\kern-1.2pt)/\varphi (a!\kern-1.2pt)\varphi (b!\kern-1.2pt)$ is an integer. We derive results on the limit of $c(a,b)/(a+b)$ as a and b tend to infinity and show that $c(a,b)>a+b$ for all pairs of positive integers $(a,b)$, with the exception of a set of density zero.

Let $\varepsilon> 0$ be sufficiently small and let $0 < \eta < 1/522$. We show that if X is large enough in terms of $\varepsilon $, then for any squarefree integer $q \leq X^{196/261-\varepsilon }$ that is $X^{\eta }$-smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod {q}$, with $(a,q) = 1$. In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096\ldots $ was replaced by $25/36 = 0.69\overline {4}$. This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$-barrier for a density 1 set of $X^{\eta }$-smooth moduli q (without the squarefree condition).

Our proof appeals to the q-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using p-adic methods.

We consider a concrete family of $2$-towers $(\mathbb {Q}(x_n))_n$ of totally real algebraic numbers for which we prove that, for each $n$, $\mathbb {Z}[x_n]$ is the ring of integers of $\mathbb {Q}(x_n)$ if and only if the constant term of the minimal polynomial of $x_n$ is square-free. We apply our characterization to produce new examples of monogenic number fields, which can be of arbitrary large degree under the ABC-Conjecture.

Using a method due to Rieger [‘Remark on a paper of Stux concerning squarefree numbers in non-linear sequences’, Pacific J. Math. 78(1) (1978), 241–242], we prove that the Piatetski-Shapiro sequence defined by $\{\lfloor n^c \rfloor : n\in \mathbb {N}\}$ contains infinitely many consecutive square-free integers whenever $1<c<3/2$.

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

Given disjoint subsets T1, …, Tm of “not too large” primes up to x, we establish that for a random integer n drawn from [1, x], the m-dimensional vector enumerating the number of prime factors of n from T1, …, Tm converges to a vector of m independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when T1, …, Tm are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors.

We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log \log n$ with the prime between $\log n$ and $(\log n)^{\log \log n}$. The proportion of even permutations with this property is at least $1-7/\log \log n$.

We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic.

We prove that neither a prime nor an l-almost prime number theorem holds in the class of regular Toeplitz subshifts. But when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler’s totient function is assumed, then the two theorems hold.

A set of integers is primitive if it does not contain an element dividing another. Let f(n) denote the number of maximum-size primitive subsets of {1,…,2n}. We prove that the limit α = limn→∞f(n)1/n exists. Furthermore, we present an algorithm approximating α with (1 + ε) multiplicative error in N(ε) steps, showing in particular that α ≈ 1.318. Our algorithm can be adapted to estimate the number of all primitive sets in {1,…,n} as well.

We address another related problem of Cameron and Erdős. They showed that the number of sets containing pairwise coprime integers in {1,…n} is between ${2^{\pi (n)}} \cdot {e^{(1/2 + o(1))\sqrt n }}$ and ${2^{\pi (n)}} \cdot {e^{(2 + o(1))\sqrt n }}$. We show that neither of these bounds is tight: there are in fact ${2^{\pi (n)}} \cdot {e^{(1 + o(1))\sqrt n }}$ such sets.