Related papers: On some problems of primes with the floor function

The function $\epsilon(x)=\mbox{li}(x)-\pi(x)$ is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions $\epsilon_{\theta}(x)=\mbox{li}[\theta(x)]-\pi(x)$ and…

Continuing previous study of the Beurling zeta function, here we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the…

Let $\big(\mathcal{S}_n^{\alpha,\kappa,\mathfrak{r}}(z)\big)_{n=1}^\infty$ be a sequence of the largest possible integer intervals, such that…

We examine oscillations in a number of sums of arithmetic functions involving $\Omega(n)$, the total number of prime factors of $n$, and $\omega(n)$, the number of distinct prime factors of $n$. In particular, we examine oscillations in…

Let $\phi(\cdot)$ and $\sigma(\cdot)$ denote the Euler function and the sum of divisors function, respectively. In this paper, we give a lower bound for the number of positive integers $m\le x$ for which the equation $m=n-\phi(n)$ has no…

For $n \in \mathbb{N}$ let $\Pi[n]$ denote the set of partitions of $n$, i.e., the set of positive integer tuples $(x_1,x_2,\ldots,x_k)$ such that $x_1 \geq x_2 \geq \cdots \geq x_k$ and $x_1 + x_2 + \cdots + x_k = n$. Fixing…

Let $f$ be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish an asymptotical formula for the quantity \[ S_f(x):=\sum_{n\leq x}\frac{f([x/n])}{[x/n]} \] as $x\rightarrow\infty$, where $[t]$ is…

Using a recent verification of the Riemann hypothesis up to height $3\cdot 10^{12}$, we provide strong estimates on $\pi(x)$ and other prime counting functions for finite ranges of $x$. In particular, we get that…

We prove that $$ \sum_{n \leq x} \varphi([x/n])\leq\bigg(\frac{1380}{4009}+\frac{2629}{4009}\cdot\frac1{\zeta(2)}+o(1)\bigg)x\log x $$ as $x\to\infty$, where $\varphi$ denotes the Euler totient function and $[x]$ denotes the integer part of…

For a set of primes $\mathcal{P}$, let $\Psi(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $\Psi(x,…

Let $\mathcal{P}$ be the set of primes and $\pi(x)$ the number of primes not exceeding $x$. Let also $P^+(n)$ be the largest prime factor of $n$ with convention $P^+(1)=1$ and $$ T_c(x)=\#\left\{p\le x:p\in \mathcal{P},P^+(p-1)\ge…

Let $\lfloor t\rfloor$ denote the integer part of $t\in\mathbb{R}$ and $\|x\|$ the distance from $x$ to the nearest integer. Suppose that $1/2<\gamma_2<\gamma_1<1$ are two fixed constants. In this paper, it is proved that, whenever $\alpha$…

Let $[\, \cdot\,]$ be the floor function. In this paper, we show that when $1<c<37/36$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c_1]+[p^c_2]+[p^c_3]\,, \end{equation*} where…

A natural number $n$ is called semi-prime if it is a product of two primes or a square of a prime. We denote $\mathbb{P}_2$ the set of all semi-primes. Our goal is to prove that for fixed integer number $a$ and sufficiently large $x$ the…

We present an improved version of the analytic method for calculating $\pi(x)$, the number of prime numbers not exceeding $x$. We implemented this method in cooperation with J. Franke, T. Kleinjung and A. Jost and calculated the value…

We study the properties of the product, which runs over the primes, $$\mathfrak{p}_n = \prod_{s_p(n) \, \geq \, p} p \quad (n \geq 1),$$ where $s_p(n)$ denotes the sum of the base-$p$ digits of $n$. One important property is the fact that…

Sophomore's dream sum $S=\sum_{n=1}^\infty n^{-n}$ is extended to the function $f(t,a)=t\int_{0}^{1}(ax)^{-tx}dx$ with $f(1,1)=S$. Asymptotic behavior for a large $|t|$ is obtained, which is exponential for $t>0$ and $t<0,a>1$, and…

Let $s$ be a fixed positive integer constant, $\varepsilon$ be a fixed small positive number. Then, provided that a prime $p$ is large enough, we prove that for any set $\{{\mathcal M}\subseteq \mathbb F_p^*$ of size $|{\mathcal M}|=…

An asymptotic formula for the sum of the first n primes is derived. This result improves the previous results of P. Dusart. Using this asymptotic expansion, we prove the conjectures of R. Mandl and G. Robin on the upper and the lower bound…

In this note, we show that there are many infinity positive integer values of $n$ in which, the following inequality holds $$ \left\lfloor{1/2}(\frac{(n+1)^2}{\log(n+1)}-\frac{n^2}{\log n})-\frac{\log^2 n}{\log\log…