query: How to prove that $\sum_{p \leq x} {\log p \over p} = \log x + O(1)$?
positive_passages.docid: 142408
positive_passages.text: How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?
negative_passages.docid: 132541
negative_passages.text: Showing ${1\over n}\sum|S_i|=O(\sqrt n)$ for $S_i\subset [n]$, $|S_i\cap S_i|\le 1$ for $i\ne j$
negative_passages.docid: 30897
negative_passages.text: $\pi(x)\geqslant\frac{\log x}{2\log2}$ for all $x\geqslant2.$
negative_passages.docid: 155232
negative_passages.text: Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
negative_passages.docid: 32510
negative_passages.text: Proving the convergence of $\sum_{n\geq 2}\log\left(1-\frac{1}{n^2}\right)$
negative_passages.docid: 121237
negative_passages.text: Solving $x - a \log(x)=b$
negative_passages.docid: 29679
negative_passages.text: Prove: $|x-y|\leq |x|+|y|$
negative_passages.docid: 3273
negative_passages.text: Trouble in proving that $\|x\|_p = \max|x_j|$
negative_passages.docid: 26833
negative_passages.text: Prove that $\mathbb P(X>Y) =\frac{b}{a + b}$ if $X, Y$ are exponentially distributed with parameters $a$ and $b$.
negative_passages.docid: 21791
negative_passages.text: Show that $\sum\limits_{k=1}^\infty \frac{1}{k(k+1)(k+2)\cdots (k+p)}=\frac{1}{p!p} $ for every positive integer $p$
negative_passages.docid: 144633
negative_passages.text: Verification of proof involving logarithms and limits
negative_passages.docid: 149889
negative_passages.text: Prove that $|\log(1 + x^2) - \log(1 + y^2)| \le |x-y|$
negative_passages.docid: 125175
negative_passages.text: convexity proof of a function including ln and sums
negative_passages.docid: 30630
negative_passages.text: Proving ${n \choose p} \equiv \Bigl[\frac{n}{p}\Bigr] \ (\text{mod} \ p)$
negative_passages.docid: 6449
negative_passages.text: Prove logarithmic inequality with greatest integer function.
negative_passages.docid: 41876
negative_passages.text: Finding an asymptotic for the sum $\sum_{p\leq x}p^m$
negative_passages.docid: 4512
negative_passages.text: Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?