The distribution of integers with at least two divisors in a short interval

We determine the order of magnitude of H(x,y,z), the number of integers n\le x having a divisor in (y,z], for all x,y and z. We also study H_r(x,y,z), the number of integers n\le x having exactly r divisors in (y,z]. When r=1 we establish the order of magnitude of H_1(x,y,z) for all x,y,z satisfying z\le x^{0.49}. For every r\ge 2, $C>1$ and $\epsilon>0$, we determine the the order of magnitude of H_r(x,y,z) when y is large and y+y/(\log y)^{\log 4 -1 - \epsilon} \le z \le \m...

Let $F(x)$ be an irreducible polynomial with integer coefficients and degree at least 2. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y<d\le z$. We determine the order of magnitude of $H_F(x, y, z)$ uniformly for $y+y/\log^C y < z\le y^2$ and $y\le x^{1-\delta}$, showing that the order is the same as the order of $H(x,y,z)$, the number of positive integers $n\le x$ with a divisor in $(y,z]...

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 integers <=x with a divisor in (y,2y]. The lower bound uses a different argument than that in the aforementioned paper. As a corollary, we deduce the order of magnitude for the number of distinct products in an N x N multiplication table.

In the late eighties, Hildebrand and Tenenbaum proved an asymptotic formula for the number of positive integers below $x$, having exactly $\nu$ distinct prime divisors: $\pi_{\nu}(x) \sim x \delta_{\nu}(x)$. Here we consider the restricted count $\pi_{\nu}(x,y)$ for integers lying in the short interval $[x,x+y]$. Extending the result of Hildebrand-Tenenbaum to this setting, we show that for any $\varepsilon >0$ \[ \pi_{\nu}(x,y) \sim y \delta_{\nu}(x), \] uniformly over all $...

We study the function $\Theta(x,y,z)$ that counts the number of positive integers $n\le x$ which have a divisor $d>z$ with the property that $p\le y$ for every prime $p$ dividing $d$. We also indicate some cryptographic applications of our results.

We present several new results involving $\Delta(x+U)-\Delta(x)$, where $U = o(x)$ and $$ \Delta(x):=\sum_{n\le x}d(n)-x\log x-(2\gamma-1)x $$ is the error term in the classical Dirichlet divisor problem.

Let \( X \geq y \geq 2 \), and let \( u = \frac{\log X}{\log y} \). We say a number is \textit{$y$-smooth} if all of its prime factors are less than or equal to \( y \). In this paper, we study the distribution of $y$-smooth numbers in short intervals. In particular, for \( y \geq \exp\left( (\log X)^{2/3 + \epsilon} \right) \), we show that the interval \( [x, x+h] \) contains a $y$-smooth number for almost all \( x \in [X, 2X] \), provided \( h \geq \exp\left( (1 + \epsilon...

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\ll (\log x)^2$. In particular we conjecture that the maximum grows surprisingly slowly as $y$ ranges from $\log x$ to $(\log x)^2$. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification.

Given a positive integer $n$ the $k$-fold divisor function $d_k(n)$ equals the number of ordered $k$-tuples of positive integers whose product equals $n$. In this article we study the variance of sums of $d_k(n)$ in short intervals and establish asymptotic formulas for the variance of sums of $d_k(n)$ in short intervals of certain lengths for $k=3$ and for $k \ge 4$ under the assumption of the Lindel\"of hypothesis.

We show that for large integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C \log_2 n$ and variance $V \log_2 n$, where $C=1/(1-e^{-\gamma})\approx 2.280$ and $V\approx 0.414$. This result is then generalized in two different directions.