Let $p(n)$ denote the least prime factor of $n$ and $L_C(x)=Cx^{1/2}(\log x)^2$ with $C>0$. The sum of $p(n)/n$ over composite $n$ lying in the short interval $[x,\,x+L_C(x)]$, a question raised by Erd\H{o}s and Graham, is studied. (i) The constant $c=8$ in the mean asymptotic is estimated \[ S(x)=\sum_{{n<x,\ n\ \text{composite}}}\frac{p(n)}{n}=\frac{c\,x^{1/2}}{(\log x)^2}\Bigl(1+O\Bigl(\frac1{\log x}\Bigr)\Bigr) +O\Bigl(\frac{x^{1/3}}{\log x}\Bigr). \] (ii)For every fixed $C>0$, the window sums $\mu_C(x):=\sum_{x\le n\le x+L_C(x)}p(n)/n$ over composites have mean $4C$: $\frac{1}{X}\sum_{x\le X}\mu_C(x)=4C+O_C(1/\log X)$, and second moment $\frac{1}{X}\sum_{x\le X}(\mu_C(x)-4C)^2=O_C((\log X)^{-2})$. In particular $\mu_C(x)=4C+o(1)$ for almost all $x$. (iii)Under a weak Cram\'{e}r-type hypothesis on primes in intervals of length $(\log y)^{2+o(1)}$, the estimate $\mu_C(x)=4C+O_C(1/\log x)$ holds uniformly in $x$, giving an affirmative answer to the Erd\H{o}s--Graham question. Unconditionally, the uniform statement remains open; proving the uniform statement unconditionally would require resolving short-interval prime estimates at scale $(\log y)^2$.
In this paper, we consider the asymptotic density of $T_c(x):=\#\{p\leq x:P^+(p-1)\geq p^c\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\rightarrow\infty$, for any $0<c\leq 1/2$, one has \begin{align*} \mathop{\lim\inf}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}&\geq\max\left(1-\frac{16}...
Let $d$ be the divisor function, and let $K(n)$ be the least positive integer $K$ for which $d((n + K)!) \ge 2d(n!)$. Erd\H{o}s, Graham, Ivi\'c and Pomerance proved that, for infinitely many $n$, \begin{equation*} K(n)>(1/9)(\log n)(\log_{2} n)(\log_{4} n)/(\log_{3} n)^3. \end{equation*} We improve upon this by a facto...
Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. We prove that \begin{align*} \sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+O_{\varepsilon,c}(x^{\theta_c+\varepsilon}), \end{align*} where $d_c$ is a suitable constant, $\theta_c<2/(3c+2)$ for $0<c...
Let $n\ge2$ be a fixed integer, and let $K,x,y$ be positive integers satisfying $2\le K\le y<x$. Consider a polynomial \[ f(u)=\alpha u^n+\alpha_{n-1}u^{n-1}+\cdots+\alpha_1u+\alpha_0 \] with real coefficients. We study the quantity \[ V_n(K,x,y) = \sum_{k=1}^{K} \left| \sum_{x-y<p\le x}e(kf(p)) \right|, \qquad e(t)=e^...
For an integer $n>1$, let $P^+(n)$ be the largest prime factor of $n$, and let $P_y^+(n)$ denote the largest prime factor of $n$ not exceeding $y$. One of Erd\H{o}s and Tur\'an's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is $1/2$. Furthermore, Wang conjectures that for...
Let $\mu$ be the M\"obius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on...
Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
