Given a positive integer m, let $\mathbb {Z}_m$ be the set of residue classes mod m. For $A\subseteq \mathbb {Z}_m$ and $n\in \mathbb {Z}_m$, let $\sigma _A(n)$ be the number of solutions to the equation $n=x+y$ with $x,y\in A$. Let $\mathcal {H}_m$ be the set of subsets $A\subseteq \mathbb {Z}_m$ such that $\sigma _A(n)\geq 1$ for all $n\in \mathbb {Z}_m$. Let

$$ \begin{align*} \ell_m=\min\limits_{A\in \mathcal{H}_m}\bigg\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}\sigma_A(n)\bigg\rbrace. \end{align*} $$

Ding and Zhao [‘A new upper bound on Ruzsa’s numbers on the Erdős–Turán conjecture’, Int. J. Number Theory 20 (2024), 1515–1523] showed that $\limsup _{m\rightarrow \infty }\ell _m\le 192$. We prove

$$ \begin{align*} \limsup\limits_{m\rightarrow\infty}\ell_m\leq 144 \end{align*} $$

and investigate parallel results on subtractive bases of $ \mathbb {Z}_m$.

We study the asymptotic behavior of the sequence $ \{\Omega (n) \}_{ n \in \mathbb {N} } $ from a dynamical point of view, where $ \Omega (n) $ denotes the number of prime factors of $ n $ counted with multiplicity. First, we show that for any non-atomic ergodic system $(X, \mathcal {B}, \mu , T)$, the operators $T^{\Omega (n)}: \mathcal {B} \to L^1(\mu )$ have the strong sweeping-out property. In particular, this implies that the pointwise ergodic theorem does not hold along $\Omega (n)$. Second, we show that the behaviors of $\Omega (n)$ captured by the prime number theorem and Erdős–Kac theorem are disjoint, in the sense that their dynamical correlations tend to zero.

We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given $\alpha \in (0,1]$ and $c>0$ (with $c\leq 1$ if $\alpha =1$), a generalized number system is constructed with Riemann prime counting function $ \Pi (x)= \operatorname {\mathrm {Li}}(x)+ O(x\exp (-c \log ^{\alpha } x ) +\log _{2}x), $ and whose integer counting function satisfies the extremal oscillation estimate $N(x)=\rho x + \Omega _{\pm }(x\exp (- c'(\log x\log _{2} x)^{\frac {\alpha }{\alpha +1}})$ for any $c'>(c(\alpha +1))^{\frac {1}{\alpha +1}}$, where $\rho>0$ is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].

Let n and k be positive integers with $n\ge k+1$ and let $\{a_i\}_{i=1}^n$ be a strictly increasing sequence of positive integers. Let $S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$ . In 1978, Borwein [‘A sum of reciprocals of least common multiples’, Canad. Math. Bull. 20 (1978), 117–118] confirmed a conjecture of Erdős by showing that $S_{n,1}\le 1-{1}/{2^{n-1}}$ . Hong [‘A sharp upper bound for the sum of reciprocals of least common multiples’, Acta Math. Hungar. 160 (2020), 360–375] improved Borwein’s upper bound to $S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$ and derived optimal upper bounds for $S_{n,2}$ and $S_{n,3}$ . In this paper, we present a sharp upper bound for $S_{n,4}$ and characterise the sequences $\{a_i\}_{i=1}^n$ for which the upper bound is attained.

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.

In this note we examine Littlewood’s proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the nonvanishing of Riemann’s zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.

Let $\text{ }\!\!\lambda\!\!\text{ }\left( n \right)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured that

*

$$\sum\limits_{n\le x}{\lambda \,\left( f\left( n \right) \right)}=o\left( x \right)$$

for any polynomial $f\left( x \right)$ with integer coefficients which is not of form $bg{{\left( x \right)}^{2}}$.

Let 〈𝒫〉⊂N be a multiplicative subsemigroup of the natural numbers N={1,2,3,…} generated by an arbitrary set 𝒫 of primes (finite or infinite). We give an elementary proof that the partial sums ∑ n∈〈𝒫〉:n≤x(μ(n))/n are bounded in magnitude by 1. With the aid of the prime number theorem, we also show that these sums converge to ∏ p∈𝒫(1−(1/p)) (the case where 𝒫 is all the primes is a well-known observation of Landau). Interestingly, this convergence holds even in the presence of nontrivial zeros and poles of the associated zeta function ζ𝒫(s)≔∏ p∈𝒫(1−(1/ps))−1 on the line {Re(s)=1}. As equivalent forms of the first inequality, we have ∣∑ n≤x:(n,P)=1(μ(n))/n∣≤1, ∣∑ n∣N:n≤x(μ(n))/n∣≤1, and ∣∑ n≤x(μ(mn))/n∣≤1 for all m,x,N,P≥1.

Let Φ(x) denote the number of those integers n with φ(n)[les ] x, where φ denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that Φ(x)-Ax [Lt] R(x), where A=ζ(2)ζ(3)/ζ(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.