Erdös and Selfridge first showed that the product of consecutive integers cannot be a perfect power. Later, this result was generalized to polynomial values by various authors. They demonstrated that the product of consecutive polynomial values cannot be the perfect power for a suitable polynomial. In this article, we consider a related problem to the product of consecutive integers. We consider all sequences of polynomial values from a given interval whose products are almost perfect powers. We study the size of these powers and give an asymptotic result. We also define a group theoretic invariant, which is a natural generalization of the Davenport constant. We provide a non-trivial upper bound of this group theoretic invariant.

We study a certain class of arithmetic functions that appeared in Klurman’s classification of $\pm 1$ multiplicative functions with bounded partial sums; c.f., Comp. Math. 153(2017), 2017, no. 8, 1622–1657. These functions are periodic and $1$-pretentious. We prove that if $f_1$ and $f_2$ belong to this class, then $\sum _{n\leq x}(f_1\ast f_2)(n)=\Omega (x^{1/4})$. This confirms a conjecture by the first author. As a byproduct of our proof, we studied the correlation between $\Delta (x)$ and $\Delta (\theta x)$, where $\theta $ is a fixed real number. We prove that there is a nontrivial correlation when $\theta $ is rational, and a decorrelation when $\theta $ is irrational. Moreover, if $\theta $ has a finite irrationality measure, then we can make it quantitative this decorrelation in terms of this measure.

We study the exact Hausdorff and packing dimensions of the prime Cantor set, $\Lambda _P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set cannot be finite and positive with respect to any sufficiently regular dimension function, thus negatively answering a question of Mauldin and Urbański (1999) and Mauldin (2013) for this class of dimension functions. By contrast, under a reasonable number-theoretic conjecture we prove that the packing measure of the conformal measure on the prime Cantor set is in fact positive and finite with respect to the dimension function $\psi (r) = r^\delta \log ^{-2\delta }\log (1/r)$, where $\delta $ is the dimension (conformal, Hausdorff, and packing) of the prime Cantor set.

Given an integer $k\ge 2$, let $\omega _k(n)$ denote the number of primes that divide n with multiplicity exactly k. We compute the density $e_{k,m}$ of those integers n for which $\omega _k(n)=m$ for every integer $m\ge 0$. We also show that the generating function $\sum _{m=0}^\infty e_{k,m}z^m$ is an entire function that can be written in the form $\prod _{p} \bigl (1+{(p-1)(z-1)}/{p^{k+1}} \bigr )$; from this representation we show how to both numerically calculate the $e_{k,m}$ to high precision and provide an asymptotic upper bound for the $e_{k,m}$. We further show how to generalize these results to all additive functions of the form $\sum _{j=2}^\infty a_j \omega _j(n)$; when $a_j=j-1$ this recovers a classical result of Rényi on the distribution of $\Omega (n)-\omega (n)$.

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.

We introduce a number field analogue of the Mertens conjecture and demonstrate its falsity for all but finitely many number fields of any given degree. We establish the existence of a logarithmic limiting distribution for the analogous Mertens function, expanding upon work of Ng. Finally, we explore properties of the generalised Mertens function of certain dicyclic number fields as consequences of Artin factorisation.

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 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].

In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are ‘close’ (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field K. For example, we show that under some conditions on rational functions $f_1, \ldots, f_n\in K(X)$, there are only finitely many elements $\alpha \in K$ such that $f_1(\alpha),\ldots,f_n(\alpha)$ are multiplicatively dependent modulo such sets.

We show that if F is $\mathbb{Q}$ or a multiquadratic number field, $p\in\left\{{2,3,5}\right\}$, and $K/F$ is a Galois extension of degree a power of p, then for elliptic curves $E/\mathbb{Q}$ ordered by height, the average dimension of the p-Selmer groups of $E/K$ is bounded. In particular, this provides a bound for the average K-rank of elliptic curves $E/\mathbb{Q}$ for such K. Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such F.

The central result is that: for each finite Galois extension $K/F$ of number fields and prime number p, as $E/\mathbb{Q}$ varies, the difference in dimension between the Galois fixed space in the p-Selmer group of $E/K$ and the p-Selmer group of $E/F$ has bounded average.

Let f be an $L^2$-normalized holomorphic newform of weight k on $\Gamma _0(N) \backslash \mathbb {H}$ with N squarefree or, more generally, on any hyperbolic surface $\Gamma \backslash \mathbb {H}$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb {Q}$. Denote by V the hyperbolic volume of said surface. We prove the sup-norm estimate

$$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$

with absolute implied constant. For a cuspidal Maaß newform $\varphi $ of eigenvalue $\lambda $ on such a surface, we prove that

$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$

We establish analogous estimates in the setting of definite quaternion algebras.

A set $S\subset {\mathbb {N}}$ is a Sidon set if all pairwise sums $s_1+s_2$ (for $s_1, s_2\in S$, $s_1\leqslant s_2$) are distinct. A set $S\subset {\mathbb {N}}$ is an asymptotic basis of order 3 if every sufficiently large integer $n$ can be written as the sum of three elements of $S$. In 1993, Erdős, Sárközy and Sós asked whether there exists a set $S$ with both properties. We answer this question in the affirmative. Our proof relies on a deep result of Sawin on the $\mathbb {F}_q[t]$-analogue of Montgomery's conjecture for convolutions of the von Mangoldt function.

For any abelian group $A$, we prove an asymptotic formula for the number of $A$-extensions $K/\mathbb {Q}$ of bounded discriminant such that the associated norm one torus $R_{K/\mathbb {Q}}^1 \mathbb {G}_m$ satisfies weak approximation. We are also able to produce new results on the Hasse norm principle and to provide new explicit values for the leading constant in some instances of Malle's conjecture.

In this paper, we investigate the asymptotic distribution of a class of multiplicative functions over arithmetic progressions without the Ramanujan conjecture. We also apply these results to some interesting arithmetic functions in automorphic context, such as coefficients of automorphic L-functions, coefficients of their Rankin–Selberg.

A positive even number is said to be a Maillet number if it can be written as the difference between two primes, and a Kronecker number if it can be written in infinitely many ways as the difference between two primes. It is believed that all even numbers are Kronecker numbers. We study the division and multiplication of Kronecker numbers and show that these numbers are rather abundant. We prove that there is a computable constant k and a set D consisting of at most 720 computable Maillet numbers such that, for any integer n, $kn$ can be expressed as a product of a Kronecker number and a Maillet number in D. We also prove that every positive rational number can be written as a ratio of two Kronecker numbers.

We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between Salié sums and to a new equidistribution estimate for the set of modular roots of primes.

We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat–Tits trees.

We use the Weyl bound for Dirichlet L-functions to derive zero-density estimates for L-functions associated to families of fixed-order Dirichlet characters. The results improve on previous bounds given by the author when $\sigma $ is sufficiently distant from the critical line.

We study bracket words, which are a far-reaching generalization of Sturmian words, along Hardy field sequences, which are a far-reaching generalization of Piatetski-Shapiro sequences $\lfloor n^c \rfloor $. We show that sequences thus obtained are deterministic (that is, they have subexponential subword complexity) and satisfy Sarnak’s conjecture.