Let $r, k, n$ be integers satisfying $1\leqslant r\leqslant k\leqslant n/2$. Let ${{\mathcal{R}}}_r(n, k)$ denote the proportion of permutations $\pi \in {{\mathcal{S}}}_n$ that fix a set of size $k$ and have no cycle of length less than $r$. In this note, we determine the order of magnitude of ${{\mathcal{R}}}_r(n, k)$ uniformly for all $2\leqslant r\leqslant k\leqslant n/2$. This result generalises the corresponding estimate of Eberhard, Ford, and Green for the case $r=1$.

Recent work by M. Afifurrahman established the first asymptotic estimates with error terms for the number of $2\times 2$ matrices with fixed non-zero determinant $n\in\mathbb{N}$, and with coefficients bounded in absolute value by X. In this paper we present a new proof of this result, which also gives an improved error term as $X\rightarrow\infty$. Similar to Afifurrahman’s result, our error term is uniform in both n and X, and our estimates are significant for X as small as $n^{1/2+\delta}$. To complement this, we also demonstrate that the exponent $1/2+\delta$ in this statement cannot be reduced, by establishing a result which gives a different asymptotic main term when n is either a prime or the square of a prime, and when $X=n^{1/2}$.

We prove new large sieve inequalities for the Fourier coefficients $\rho _{j\mathfrak {a}}(n)$ of exceptional Maass forms of a given level, weighted by sequences $(a_n)$ with sparse Fourier transforms – including two key types of sequences that arise in the dispersion method. These give the first savings in the exceptional spectrum for the critical case of sequences as long as the level, and lead to improved bounds for various multilinear forms of Kloosterman sums. As an application, we show that the greatest prime factor of $n^2+1$ is infinitely often greater than $n^{1.3}$, improving Merikoski’s previous threshold of $n^{1.279}$. We also announce applications to the exponents of distribution of primes and smooth numbers in arithmetic progressions.

In 1970, Apostol introduced the Möbius function of order k for an integer $k\ge 1$. In 2001, it was generalized by Bege to the Möbius function $\mu _{k,m}$ of two parameters for any integers $m\ge k\ge 1$. In this article, we will establish three kinds of asymptotic formulas related to $\mu _{k,m}$ for $m\ge k\ge 2$ in a unified elementary method. These formulas are related to the shifted convolution sums of Fourier coefficients of holomorphic cusp forms, Bergelson and Richter’s dynamical generalization of the prime number theorem, and the Titchmarsh divisor problem.

Let $\omega (n)$ denote the number of distinct prime factors of a natural number n. In 1940, Erdős and Kac established that $\omega (n)$ obeys the Gaussian distribution over natural numbers, and in 2004, the third author generalized their theorem to all abelian monoids. In this article, we extend her theorem to any subsets of an abelian monoid satisfying some additional conditions, and apply this result to the subsets of h-free and h-full elements. We study generalizations of several arithmetic functions, such as the prime counting omega functions and the divisor function in a unified framework. Finally, we apply our results to number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad relevance of our approach.

We prove that the average size of a mixed character sum

\begin{equation*}\sum_{1\leqslant n \leqslant x} \chi(n) e(n\theta) w(n/x)\end{equation*}

$w$

$\sqrt{x}$

$\theta$

$\chi$

$r$

$x\leqslant r$

$o(\sqrt{x})$

$\theta$

Turán observed that logarithmic partial sums $\sum_{n\le x}{f(n)}/{n}$ of completely multiplicative functions (in the particular case of the Liouville function $f(n)=\lambda(n)$) tend to be positive. We develop a general approach to prove two results aiming to explain this phenomena.

Firstly, we show that there exist constants $C, x_0\ge 1,$ such that for any completely multiplicative function f satisfying $-1\le f(n)\le 1$, we have

\begin{equation*}\sum_{n\le x}\frac{f(n)}{n}\ge -\frac{C(\log\log\log{x})^2}{(\log\log{x})}, \quad x\ge x_0.\end{equation*}

$f:\mathbb{N}\to \{-1,1\}$

$\sum_{n\le x}{f(n)}/{n}$

$O(\exp(-\exp({\log x\cdot \log\log\log x}/{C\log \log x}))).$

Recently, Donoso, Le, Moreira, and Sun studied the asymptotic behaviour of the averages of completely multiplicative functions over the Gaussian integers. They derived Wirsing’s theorem for Gaussian integers, answered a question of Frantzikinakis and Host for the sum of two squares, and obtained a variant of a theorem of Bergelson and Richter on ergodic averages along the number of prime factors of integers. In this paper, we will show the analogue of these results for co-prime integer pairs. Moreover, building on Frantzikinakis and Host’s results, we obtain some convergences on the multilinear averages of multiplicative functions over primitive lattice points.

In this paper, we introduce a new technique to study the distribution in residue classes of sets of integers with digit and sum-of-digits restrictions. From our main theorem, we derive a necessary and sufficient condition for integers with missing digits to be uniformly distributed in arithmetic progressions, extending previous results going back to the work of Erdős, Mauduit and Sárközy. Our approach uses Markov chains and does not rely on Fourier analysis as many results of this nature do. Our results apply more generally to the class of multiplicatively invariant sets of integers. This class, defined by Glasscock, Moreira and Richter using symbolic dynamics, is an integer analogue to fractal sets and includes all missing digits sets. We address uniform distribution in this setting, partially answering an open question posed by the same authors.

Recently, there has been a large number of works on bilinear sums with Kloosterman sums and on sums of Kloosterman sums twisted by arithmetic functions. Motivated by these, we consider several related new questions about sums of Kloosterman sums parametrised by square-free and smooth integers. Some of our results are presented in the much more general setting of trace functions.

Martin, Mossinghoff, and Trudgian [19] recently introduced a family of arithmetic functions called “fake $\mu $’s,” which are multiplicative functions for which there is a $\{-1,0,1\}$-valued sequence $(\varepsilon _j)_{j=1}^{\infty }$ such that $f(p^j) = \varepsilon _j$ for all primes p. They investigated comparative number-theoretic results for fake $\mu $’s and, in particular, proved oscillation results at scale $\sqrt {x}$ for the summatory functions of fake $\mu $’s with $\varepsilon _1=-1$ and $\varepsilon _2=1$. In this article, we establish new oscillation results for the summatory functions of all nontrivial fake $\mu $’s at scales $x^{1/2\ell }$ where $\ell $ is a positive integer (the “critical index”) depending on f; for $\ell =1$ this recovers the oscillation results in [19]. Our work also recovers results on the indicator functions of powerfree and powerfull numbers; we generalize techniques applied to each of these examples to extend to all fake $\mu $’s.

We provide new upper bounds for sums of certain arithmetic functions in many variables at polynomial arguments and, exploiting recent progress on the mean-value of the Erdős—Hooley $\Delta$-function, we derive lower bounds for the cardinality of those integers not exceeding a given limit that are expressible as certain sums of powers.

We show that smooth numbers are equidistributed in arithmetic progressions to moduli of size $x^{66/107-o(1)}$. This overcomes a longstanding barrier of $x^{3/5-o(1)}$ present in previous works of Bombieri, Friedlander and Iwaniec, Fouvry and Tenenbaum, Drappeau, and Maynard. We build on Drappeau’s variation of Linnik’s dispersion method and on exponential sum manipulations of Maynard, ultimately relying on optimized Deshouillers–Iwaniec-type estimates for sums of Kloosterman sums.

In the early 2000s, Ramakrishna asked the question: for the elliptic curve

\[E\;:\; y^2 = x^3 - x,\]

$a_p(E)$

$E/\mathbb{Q}$

Brun’s constant is the summation of the reciprocals of all twin primes, given by

$$ \begin{align*}B=\sum_{p \in P_2}{\bigg( \frac{1}{p} + \frac{1}{p+2}\bigg)}.\end{align*} $$

While rigorous unconditional bounds on B are known, we present the first rigorous bound on Brun’s constant under the assumption of GRH, yielding $B < 2.1594$.

Weighted sieves are used to detect numbers with at most S prime factors with $S \in \mathbb{N}$ as small as possible. When one studies problems with two variables in somewhat symmetric roles (such as Chen primes, that is primes p such that $p+2$ has at most two prime factors), one can utilise the switching principle. Here we discuss how different sieve weights work in such a situation, concentrating in particular on detecting a prime along with a product of at most three primes.

As applications, we improve on the works of Yang and Harman concerning Diophantine approximation with a prime and an almost prime, and prove that, in general, one can find a pair $(p, P_3)$ when both the original and the switched problem have level of distribution at least $0.267$.

We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus q, any two reduced congruence classes $a_1$ and $a_2$ mod q, and any $r_1,r_2 \ge 1$, a positive density of sums of two squares begin a chain of $r_1$ consecutive sums of two squares, all of which are $a_1$ mod q, followed immediately by a chain of $r_2$ consecutive sums of two squares, all of which are $a_2$ mod q. This is an analog of the result of Maynard for the sequence of primes, showing that for any reduced congruence class a mod q and for any $r \ge 1$, a positive density of primes begin a sequence of r consecutive primes, all of which are a mod q.

Let $k{\geqslant} 2$ be an integer and let $\lambda$ be the Liouville function. Given k non-negative distinct integers $h_1,\ldots,h_k$, the Chowla conjecture claims that $\sum_{n{\leqslant} x}\lambda(n+h_1)\cdots \lambda(n+h_k)=o(x)$. An unconditional answer to this conjecture is yet to be found, and in this paper, we take a conditional approach. More precisely, we establish a non-trivial bound for the sums $\sum_{n{\leqslant} x}\lambda(n+h_1)\cdots \lambda(n+h_k)$ under the existence of a Landau–Siegel zero for x in an interval that depends on the modulus of the character whose Dirichlet series corresponds to the Landau–Siegel zero. Our work constitutes an improvement over the previous related results of Germán and Kátai, Chinis and Tao and Teräväinen.

A square-full number is a positive integer for which all prime divisors divide the number at least twice. For a relatively prime pair $(a,b)\in \mathbb N\times \mathbb N\cup \{0\}$ and an affine polynomial $f(x)=ax+b$, the number of $n\leqslant N$ such that $f(n)$ is square-full is of order $N^{{1}/{2}}$. For $\varepsilon>0$ and an admissible quadratic polynomial $f(x)\in \mathbb Z[x]$, we show that the number of $n\leqslant N$ such that $f(n)$ is square-full is at most $O_{\varepsilon ,f}(N^{\varpi +\varepsilon })$ for some absolute constant $\varpi <1/2$. Under the $abc$ conjecture, we expect the upper bound to be $O_{\varepsilon ,f}(N^\varepsilon )$.