arXiv subjects
Publications and source records attributed to S. I. Dimitrov.
This work is concerned with the study of lower bounds for various expressions related to the arithmetic functions $φ(n)$, $ψ(n)$ and $σ(n)$. Several explicit estimates are established.
Let $[\, x\,]$ denote the integer part of a real number $x$. Assume that $λ_1,λ_2,λ_3$ are nonzero real numbers, not all of the same sign, that $λ_1/λ_2$ is irrational, and that $η$ is real. Let $\frac{219}{220}<γ<1$ and $θ>0$. We establish that, there exist infinitely many triples of primes $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p^4_3+η|<\big(\max \{p_1, p_2, p^4_3\}\big)^{\frac{219-220γ}{208}+θ} \end{equation*} and such that $p_i=[n_i^{1/γ}]$, $i=1,\,2,\,3$.
Let $[\,\cdot\,]$ denote the floor function. In this paper, we show that whenever $η$ is real and the constants $λ_i$ satisfy some necessary conditions, then for any fixed $\frac{63}{64}<γ<1$ and $θ>0$, there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p^2_3+η|<\big(\max \{p_1, p_2, p^2_3\}\big)^{{\frac{63-64γ}{52}}+θ} \end{equation*} and such that $p_i=[n_i^{1/γ}]$, $i=1,\,2,\,3$.
Let $[\,\cdot\,]$ denote the floor function. Assume that $λ_1, λ_2, λ_3, λ_4, λ_5$ are nonzero real numbers, not all of the same sign, that $λ_1/λ_2$ is irrational, and that $η$ is a real number. Let $\frac{71}{72}<γ<1$ and $θ>0$. We prove that there exist infinitely many quintuples of primes $p_1,\, p_2,\, p_3,\, p_4,\, p_5$ satisfying the Diophantine inequality \begin{equation*} \big|λ_1p^2_1 + λ_2p^2_2 + λ_3p^2_3+ λ_4p^2_4 + λ_5p^2_5+η\big|<\big(\max p_j\big)^{\frac{71-72γ}{29}+θ}\,, \end{equation*} where $p_i=[n_i^{1/γ}]$, $i=1,\,2,\,3,\,4,\,5$. We also prove analogous theorems by raising the last variable in the inequality to the third and fourth powers.
In this article, we study the properties of $ψ$-amicable numbers. We prove that their asymptotic density relative to the positive integers is zero. We also propose generalizations of $ψ$-amicable numbers.
Let $[\, \cdot\,]$ be the floor function. In this paper, we show that when $1<c<\frac{82}{79}$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c]+[m^c]\,, \end{equation*} where $p$ is a prime and $m$ is a square-free.
In this paper, we introduce the notion of $ψ$-quadratic $k$-tuples. We also give examples, prove some properties and propose generalizations of these new concepts.
In this paper, we propose new generalizations of amicable numbers. We also give examples and prove properties of these new concepts.
Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denote the distance from $x$ to the nearest integer. In this paper we show that whenever $α$ is irrational and $β$ is real then for any fixed $\frac{13}{14}<γ<1$, there exist infinitely many prime numbers $p$ satisfying the inequality \begin{equation*} \|αp^2+β\|< p^{\frac{13-14γ}{29}+\varepsilon} \end{equation*} and such that $p=[n^{1/γ}]$.
Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denotes the distance from $x$ to the nearest integer. In this paper we show that whenever $α$ is irrational and $β$ is real then for any fixed $1<c<12/11$ there exist infinitely many prime numbers $p$ satisfying the inequality \begin{equation*} \|αp+β\|\ll p^{\frac{11c-12}{26c}}\log^6p \end{equation*} and such that $p=[n^c]$.
In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed $1 0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3$, such that $p_1=x^2 + y^2 +1$. For this purpose we establish a new Bombieri -- Vinogradov type result for exponential sums over primes.
In this paper, we establish a Bombieri-Vinogradov type result for prime numbers of the form $p=x^2+y^2+1$. The proof is based on the enveloping sieve.
In this paper we establish three Barban-Davenport-Halberstam type theorems. Namely for exponential sums over primes, for Piatetski-Shapiro primes and for exponential sums over Piatetski-Shapiro primes.
In this paper we establish lower bounds on several expressions dependent on functions $φ(n)$, $ψ(n)$ and $σ(n)$.
This paper presents a brief survey of the most important and the most remarkable inequalities involving the basic arithmetic functions.
Let $[\, \cdot\,]$ be the floor function. In this paper, we prove that when $1<c<\frac{16559}{15276}$, 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 $p_1,p_2,p_3$ are primes, such that $p_1=x^2 + y^2 +1$.
In this paper we show that, for any fixed $1 0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c+p_5^c-N|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3,\,p_4,\,p_5$, such that $p_1=x^2 + y^2 +1$.
In this paper we show that, for any fixed $1 0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3,\,p_4$, such that $p_1=x^2 + y^2 +1$.