Let f(x) and g(x) be polynomials in $\mathbb F_{2}[x]$ with ${\rm deg}\text{ } f=n$. It is shown that for $n\gg 1$, there is an $g_{1}(x)\in \mathbb F_{2}[x]$ with ${\rm deg}\text{ } g_{1}\leqslant \max\{{\rm deg}\text{ } g, 6.7\log n\}$ and $g(x)-g_{1}(x)$ having $ \lt 6.7\log n$ terms such that $\gcd(f(x), g_{1}(x))=1$. As an application, it is established using a result of Dubickas and Sha that given $f(x)\in \mathbb F_{2}[x]$ of degree $n\geqslant 1$, there is a separable $g(x)\in 2[x]$ with ${\rm deg}\text{ } g= {\rm deg}\text{ } f$ and satisfying that $f(x)-g(x)$ has $\leqslant 6.7\log n$ terms. As a simple consequence, the latter result holds in $\mathbb Z[x]$ after replacing ‘number of terms’ by the L1-norm of a polynomial and $6.7\log n$ by $6.8\log n$. This improves the bound $(\log n)^{\log 4 +\operatorname{\varepsilon}}$ obtained by Filaseta and Moy.

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

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

For $k\geq 2$ and a nonzero integer n, a generalised Diophantine m-tuple with property $D_k(n)$ is a set of m positive integers $S = \{a_1,a_2,\ldots , a_m\}$ such that $a_ia_j + n$ is a kth power for $1\leq i< j\leq m$. Define $M_k(n):= \text {sup}\{|S| : S$ having property $D_k(n)\}$. Dixit et al. [‘Generalised Diophantine m-tuples’, Proc. Amer. Math. Soc. 150(4) (2022), 1455–1465] proved that $M_k(n)=O(\log n)$, for a fixed k, as n varies. In this paper, we obtain effective upper bounds on $M_k(n)$. In particular, we show that for $k\geq 2$, $M_k(n) \leq 3\,\phi (k) \log n$ if n is sufficiently large compared to k.

Let $f(X) \in {\mathbb Z}[X]$ be a polynomial of degree $d \ge 2$ without multiple roots and let ${\mathcal F}(N)$ be the set of Farey fractions of order N. We use bounds for some new character sums and the square-sieve to obtain upper bounds, pointwise and on average, on the number of fields ${\mathbb Q}(\sqrt {f(r)})$ for $r\in {\mathcal F}(N)$, with a given discriminant.

Let f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let $\{\lambda_f(n)\}_n$ be its sequence of normalised Fourier coefficients. We show that if $K/ \mathbb{Q}$ is any number field, and $\mathcal{N}_K$ denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers $n \in \mathcal{N}_K$ yield a sign change for the sequence $\{\lambda_f(n)\}_{n \in \mathcal{N}_K}$. More precisely, for a positive proportion of $n \in \mathcal{N}_K \cap [1,X]$ we have $\lambda_f(n)\lambda_f(n') < 0$, where n′ is the first element of $\mathcal{N}_K$ greater than n for which $\lambda_f(n') \neq 0$.

For example, for $K = \mathbb{Q}(i)$ and $\mathcal{N}_K = \{m^2+n^2 \;:\; m,n \in \mathbb{Z}\}$ the set of sums of two squares, we obtain $\gg_f X/\sqrt{\log X}$ such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwiłł on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author.

In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed $a \neq 0$ there are $\gg_{f,\varepsilon} X^{1/2-\varepsilon}$ sign changes for $\lambda_f$ along the sequence of integers of the form $a + m^2 + n^2 \leq X$.

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 $\varepsilon>0$ .

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.

In this article we establish the arithmetic purity of strong approximation for certain semisimple simply connected linear algebraic groups and their homogeneous spaces over a number field $k$. For instance, for any such group $G$ and for any open subset $U$ of $G$ with ${\mathrm {codim}}(G\setminus U, G)\geqslant 2$, we prove that (i) if $G$ is $k$-simple and $k$-isotropic, then $U$ satisfies strong approximation off any finite number of places; and (ii) if $G$ is the spin group of a non-degenerate quadratic form which is not compact over archimedean places, then $U$ satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of $G\setminus U$, and an affine linear sieve which allows us to produce integral points with almost-prime polynomial values.

In this paper we sharpen Hildebrand’s earlier result on a conjecture of Erdős on limit points of the sequence ${\{d(n)/d(n+1)\}}$ .

For each positive integer n, let $U(\mathbf {Z}/n\mathbf {Z})$ denote the group of units modulo n, which has order $\phi (n)$ (Euler’s function) and exponent $\lambda (n)$ (Carmichael’s function). The ratio $\phi (n)/\lambda (n)$ is always an integer, and a prime p divides this ratio precisely when the (unique) Sylow p-subgroup of $U(\mathbf {Z}/n\mathbf {Z})$ is noncyclic. Write W(n) for the number of such primes p. Banks, Luca, and Shparlinski showed that for certain constants $C_1, C_2>0$,

$$ \begin{align*} C_1 \frac{\log\log{n}}{(\log\log\log{n})^2} \le W(n) \le C_2 \log\log{n} \end{align*} $$

$\log \log {n}/\log \log \log {n}$

This paper is concerned with the function r3(n), the number of representations of n as the sum of at most three positive cubes,

$$r_3(n) = {\mathrm{card}}\{\mathbf m\in\mathbb Z^3: m_1^3+m_2^3+m_3^3=n, m_j\ge1\}.$$

$${\sum_{m=1}^nr_3(m),\,\sum_{m=1}^nr_3(m)^2}$$

$${\sum_{\substack{ n\le x\\ n\equiv a\,\mathrm{mod}\,q }} r_3(n).\}$$

We investigate the density of square-free values of polynomials with large coefficients over the rational function field 𝔽q[t]. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial N as a sum of a k-th power of a small polynomial and a square-free polynomial.

Given a positive integer n let ω (n) denote the number of distinct prime factors of n, and let a be fixed positive integer. Extending work of Kubilius, we develop a bivariate probabilistic model to study the joint distribution of the deterministic vectors (ω(n), ω(n + a)), with n ≤ x as x → ∞, where n and n + a belong to a subset of ℕ with suitable properties. We thus establish a quantitative version of a bivariate analogue of the Erdős–Kac theorem on proper subsets of ℕ.

We give three applications of this result. First, if y = x0(1) is not too small then we prove (in a quantitative way) that the y-truncated Möbius function μy has small binary autocorrelations. This gives a new proof of a result due to Daboussi and Sarkőzy. Second, if μ(n; u) :=e(uω(n)), where u ∈ ℝ then we show that μ(.; u) also has small binary autocorrelations whenever u = o(1) and $u\sqrt {\mathop {\log }\nolimits_2 x} \to \infty$, as x → ∞. These can be viewed as partial results in the direction of a conjecture of Chowla on binary correlations of the Möbius function.

Our final application is related to a problem of Erdős and Mirsky on the number of consecutive integers less than x with the same number of divisors. If $y = x^{{1 \over \beta }}$, where β = β(x) satisfies certain mild growth conditions, we prove a lower bound for the number of consecutive integers n ≤ x that have the same number of y-smooth divisors. Our bound matches the order of magnitude of the one conjectured for the original Erdős-Mirsky problem.

Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field ${{\mathbb{F}}_{p}}$, we study the frequency to which a given group $G$ arises as a group of points $E\left( {{\mathbb{F}}_{p}} \right)$. It is well known that the only permissible groups are of the form ${{G}_{m,\,k}}\,:=\,\mathbb{Z}\,/m\mathbb{Z}\,\times \,\mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M\left( {{G}_{m,\,k}} \right)$ be the frequency to which the group ${{G}_{m,\,k}}$ arises in this way. Previously, C.David and E. Smith determined an asymptotic formula for $M\left( {{G}_{m,\,k}} \right)$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M\left( {{G}_{m,\,k}} \right)$, pointwise and on average. In particular, we show that $M\left( {{G}_{m,\,k}} \right)$ is bounded above by a constant multiple of the expected quantity when $m\,\le \,{{k}^{A}}$ and that the conjectured asymptotic for $M\left( {{G}_{m,\,k}} \right)$ holds for almost all groups ${{G}_{m,\,k}}$ when $m\,\le \,{{k}^{1/4-\in }}$. We also apply our methods to study the frequency to which a given integer $N$ arises as a group order $\#E\left( {{\mathbb{F}}_{p}} \right)$.

Given a family of varieties $X\rightarrow \mathbb{P}^{n}$ over a number field, we determine conditions under which there is a Brauer–Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.

Let $E(N)$ denote the number of positive integers $n\leqslant N$, with $n\equiv 4\;(\text{mod}\;24)$, which cannot be represented as the sum of four squares of primes. We establish that $E(N)\ll N^{11/32}$, thus improving on an earlier result of Harman and the first author, where the exponent $7/20$ appears in place of $11/32$.

Let $A\subset \{1,\dots ,N\}$ be a set of prime numbers containing no non-trivial arithmetic progressions. Suppose that $A$ has relative density ${\it\alpha}=|A|/{\it\pi}(N)$, where ${\it\pi}(N)$ denotes the number of primes in the set $\{1,\dots ,N\}$. By modifying Helfgott and De Roton’s work [Improving Roth’s theorem in the primes. Int. Math. Res. Not. IMRN2011 (4) (2011), 767–783], we improve their bound and show that

$$\begin{eqnarray}{\it\alpha}\ll \frac{(\log \log \log N)^{6}}{\log \log N}.\end{eqnarray}$$

We give an upper bound for the number of elliptic Carmichael numbers $n\,\le \,x$ that were recently introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non $\text{CM}$). We also discuss several possible further improvements.

We investigate sums of mixed powers involving two squares and three biquadrates. In particular, subject to the truth of the Generalised Riemann Hypothesis and the Elliott–Halberstam conjecture, we show that all large natural numbers $n$ with $8\nmid n$, $n\not\equiv 2~(\text{mod} ~3)$ and $n\not\equiv 14~(\text{mod} ~16)$ are the sum of two squares and three biquadrates.