Related papers: Divisors of shifted primes

Let $N(x,y)$ denote the number of integers $n\le x$ which are divisible by a shifted prime $p-1$ with $p>y$, $p$ prime. Improving upon recent bounds of McNew, Pollack and Pomerance, we establish the exact order of growth of $N(x,y)$ for all…

We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set…

Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, every reduced arithmetic progression $a\mod q$ has approximately the expected number of primes $p$ from the interval…

We give a relatively short proof of one of the central cases of the main theorem from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223. Namely, we determine the order of magnitude of the number of…

Let $a,b\in\mathbb{Z}\setminus\{0\}$. For every $n\in\mathbb{N}$, denote by $\omega_a^*(n)$ the number of shifted-prime divisors $p-a$ of $n$, where $p>a$ is prime. In this paper, we study the moments of $\omega_a^*$ over shifted primes…

Let $Q$ be a set of primes with relative density $\delta$. We count integers in $[1,x]$ with prime factors all in $Q$ that also have a divisor in $(y,2y]$. We establish the order of magnitude for all $\delta \in (0,1]$. This generalizes the…

Let $\omega^*(n)$ denote the number of divisors of $n$ that are shifted primes, that is, the number of divisors of $n$ of the form $p-1$, with $p$ prime. Studied by Prachar in an influential paper from 70 years ago, the higher moments of…

We develop an analog for shifted primes of the Kubilius model of prime factors of integers. We prove a total variation distance estimate for the difference between the model and actual prime factors of shifted primes, and apply it to show…

Best possible bounds are obtained for the concentration function of an additive arithmetic function on sequences of shifted primes.

Let $\tau(n)$ stand for the number of divisors of the positive integer $n$. We obtain upper bounds for $\tau(n)$ in terms of $\log n$ and the number of distinct prime factors of $n$.

We prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a…

We obtain a lower bound for \[ \#\{x/2< p_{n}\leq x:\ p_n \equiv\ldots\equiv p_{n+m}\equiv a\text{ (mod $q$)},\ p_{n+m} - p_{n}\leq y\}, \] where $p_{n}$ is the $n^{\text{th}}$ prime.

We obtain an asymptotic formula for the number of primes $p\leq x_1$, $p\leq x_2$ such that $p_1(p_2+a)\equiv l \pmod q$ with $(a,q)=(l,q)=1$, $q\leq x^{\kappa_0}$, $x_1\geq x^{1-\alpha}$, $x_2\geq x^{\alpha}$, $$…

In this paper we propose necessary upper bounds for the second, third and fourth smallest prime divisors of friends of 10 based on the number of distinct prime divisors of it.

We establish an analog of the Hardy-Ramanujan inequality for counting members of sifted sets with a given number of distinct prime factors. In particular, we establish a bound for the number of shifted primes p+a below x with k distinct…

We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_{\nu}x^{n_\nu}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_\nu$ and their differences. In…

We consider the distribution of the largest prime divisor of the integers in the interval $[2,x]$, and investigate in particular the mode of this distribution, the prime number(s) which show up most often in this list. In addition to giving…

We obtain an upper bound for the distribution of primes in the form $n^4 + k$ up to $x$, averaged over $k$ with small square-full part. As a corollary, we show that for almost all $k$, there is an expected amount of primes in the form $n^4…

Let $x$ and $n$ be positive integers. We prove a non-trivial lower bound for $x$, dependant only on $\omega_n$, the number of distinct prime factors of $x^n-1$. By considering the divisibility of $\varphi \mid x^n-1$ for $\varphi \mid n$,…

We determine the order of magnitude of H^{(k+1)}(x,\vec{y},2\vec{y}), the number of integers up to x that are divisible by a product d_1...d_k with y_i<d_i\le 2y_i, when the numbers \log y_1,...,\log y_k have the same order of magnitude and…