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 $a + m^2 + n^2 \leq X$.

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.

We investigate the leading digit distribution of the kth largest prime factor of n (for each fixed $k=1,2,3,\dots $ ) as well as the sum of all prime factors of n. In each case, we find that the leading digits are distributed according to Benford’s law. Moreover, Benford behavior emerges simultaneously with equidistribution in arithmetic progressions uniformly to small moduli.

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 $1<c<3/2$ .

Let $k$ be an arbitrary positive integer and let $\unicode[STIX]{x1D6FE}(n)$ stand for the product of the distinct prime factors of $n$. For each integer $n\geqslant 2$, let $a_{n}$ and $b_{n}$ stand respectively for the maximum and the minimum of the $k$ integers $\unicode[STIX]{x1D6FE}(n+1),\unicode[STIX]{x1D6FE}(n+2),\ldots ,\unicode[STIX]{x1D6FE}(n+k)$. We show that $\liminf _{n\rightarrow \infty }a_{n}/b_{n}=1$. We also prove that the same result holds in the case of the Euler function and the sum of the divisors function, as well as the functions $\unicode[STIX]{x1D714}(n)$ and $\unicode[STIX]{x1D6FA}(n)$, which stand respectively for the number of distinct prime factors of $n$ and the total number of prime factors of $n$ counting their multiplicity.

Using elementary means, we improve an explicit bound on the divisor function due to Friedlander and Iwaniec [Opera de Cribro, American Mathematical Society, Providence, RI, 2010]. Consequently, we modestly improve a result regarding a sieving inequality for Gaussian sequences.

Estimating averages of Dirichlet convolutions $1\ast \unicode[STIX]{x1D712}$, for some real Dirichlet character $\unicode[STIX]{x1D712}$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin’s conjecture for Châtelet surfaces. We introduce a far-reaching generalisation of this problem, in particular replacing $\unicode[STIX]{x1D712}$ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin’s conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number.

The multiplicative group generated by a certain sequence of rationals is determined, settling a 30-year conjecture.

For natural integer n, let D n denote the random variable taking the values log d for d dividing n with uniform probability 1/τ(n). Then t↦ℙ(D n ≤n t ) (0≤t≤1) is an arithmetic process with respect to the uniform probability over the first N integers. It is known from previous works that this process converges to a limit law and that the same holds for various extensions. We investigate the generalized moments of arbitrary orders for the limit laws. We also evaluate the mean value of the two-dimensional distribution function ℙ(D n ≤n u , D{n/D n}≤n v ).

Let ${\it\lambda}$ and ${\it\mu}$ denote the Liouville and Möbius functions, respectively. Hildebrand showed that all eight possible sign patterns for $({\it\lambda}(n),{\it\lambda}(n+1),{\it\lambda}(n+2))$ occur infinitely often. By using the recent result of the first two authors on mean values of multiplicative functions in short intervals, we strengthen Hildebrand’s result by proving that each of these eight sign patterns occur with positive lower natural density. We also obtain an analogous result for the nine possible sign patterns for $({\it\mu}(n),{\it\mu}(n+1))$ . A new feature in the latter argument is the need to demonstrate that a certain random graph is almost surely connected.

We investigate the behaviour of the function $L_{\alpha }(x) = \sum _{n\leq x}\lambda (n)/n^{\alpha }$, where $\lambda (n)$ is the Liouville function and $\alpha $ is a real parameter. The case where $\alpha =0$ was investigated by Pólya; the case $\alpha =1$, by Turán. The question of the existence of sign changes in both of these cases is related to the Riemann hypothesis. Using both analytic and computational methods, we investigate similar problems for the more general family $L_{\alpha }(x)$, where $0\leq \alpha \leq 1$, and their relationship to the Riemann hypothesis and other properties of the zeros of the Riemann zeta function. The case where $\alpha =1/2$is of particular interest.

Let P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n > 1. We show that, for every real number , there exists a constant c(η) > 1 such that for every integer a ≠ 0, the set has relative asymptotic density one in the set of all prime numbers. Moreover, in the range , one can take c(η) = 1+ε for any fixed ε > 0. In particular, our results imply that for every real number 0.486 ≤ b.thetav; ≤ 0.531, the relation P(q − a) ≍ qθ holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime divisor of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q ↦ P(q - a) for a > 0, and show that for infinitely many primes q, this map can be iterated at least (log logq)1+o(1) times before it terminates.

We consider the weak convergence of the set of strongly additive functions f(q) with rational argument q. It is assumed that f(p) and f(1/p) ∈ {0, 1} for all primes. We obtain necessary and sufficient conditions of the convergence to the limit distribution. The proof is based on the method of factorial moments. Sieve results, and Halász's and Ruzsa's inequalities are used. We present a few examples of application of the given results to some sets of fractions.

In 1989, Alladi, Erdös and Vaaler confirmed their own conjecture about a class of multiplicative functions by means of a deep result of Baranyai on hypergraphs. In this note we give a simple direct proof of the result which is derived in their proof as a consequence of the above mentioned graph theoretic result.

We investigate the Ω±-result of Δa(x) and its number of sign-changes in an interval [1, T], denoted by Xa(T), for -1 ≤ a < -1/2. We can prove that T ≪aXa(T) which is the best possible in order of magnitude.

We give a discussion on the properties of Δa(x) (− 1 < a < 0), which is a generalization of the error term Δ(x) in the Dirichlet divisor problem. In particular, we study its oscillatory nature and investigate the gaps between its sign-changes for −½ ≤ a < 0.