The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.

Let $\left[t\right]$ be the integral part of a real number $t$, and let $f$ be the arithmetic function satisfying some simple condition. We establish a new asymptotical formula for the sum $S_f\left(x\right)={\sum }_{n\le x}f\left([x/n]\right)$, which improves the recent result of J. Stucky (2022).

The paper deals with asymptotics for a class of arithmetic functions which describe the value distribution of the greatest-common-divisor function. Typically, they are generated by a Dirichlet series whose analytic behavior is determined by the factor ζ2(s)ζ(2s − 1). Furthermore, multivariate generalizations are considered.

We study the average of the Fourier coefficients of a holomorphic cusp form for the full modular group at primes of the form [g(n)].

We consider $k$-free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number $\alpha >1$ of finite type $\tau <\infty$ and any constant $\epsilon >0$, we can show that $$\sum _{\genfrac{}{}{0pt}{}{1\le n\le x}{[\alpha n+\beta ]\in {𝒬}_k}}1-\frac{x}{\zeta \left(k\right)}\ll x^{k/(2k-1)+\epsilon }+x^{1-1/(\tau +1)+\epsilon },$$ where ${𝒬}_k$ is the set of positive $k$-free integers and the implied constant depends only on $\alpha ,$$\epsilon ,$$k$...

Linnik proved, assuming the Riemann Hypothesis, that for any $ϵ\&gt;0$, the interval $[N,N+{log}^{3+ϵ}N]$ contains a number which is the sum of two primes, provided that $N$ is sufficiently large. This has subsequently been improved to the same assertion being valid for the smaller gap $C{log}^{2}N$, the added new ingredient being Selberg’s estimate for the mean-square of primes in short intervals. Here we give another proof of this sharper result which avoids the use of Selberg’s estimate and is therefore more in the spirit of Linnik’s...

The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$, by performing sum and product set operations starting from a given subset $A$ of ${\mathbb{Z}}_q$. We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $\left|A\right|\sim logq$). Roughly speaking we show that all residue classes are obtained from a $k$-fold sum of an $r$-fold product set of $A$, where $r\ll logq$ and $logk\ll logq$, provided the residue sets...