Sorry about the title i didn't know what to call this other than "analytic number theory"  I am asked to show that $\pi (x) = -1 + \pi(\sqrt x) + \sum \mu (d)\lfloor \frac {x}{d}\rfloor$, where the sum is over all d for which ever prime factor is less or equal to $\sqrt x $  The furthest right hand part of the equation looks like a mobius inversion but im not sure how to do it over root x and the rest i am completely lost on.
Legendre's formula counts the number of positive integers less than or equal to a number $x$ which are not divisible by any of the first $a$ primes: $$\begin{align} &amp;\phi(x,a)=\lfloor x \rfloor-\sum_{p_i\le a}\left\lfloor \dfrac{ x }{(p_i)}\right\rfloor+\sum_{p_i&lt;p_j\le a}\left\lfloor\dfrac{ x}{(p_ip_j)}\right\rfloor-\sum_{p_i&lt;p_j&lt;p_k\le a}\left\lfloor \dfrac{x}{(p_ip_jp_k)}\right\rfloor+\dots \end{align}$$ then $\pi(x)=\phi\bigl(x,\pi (\sqrt{x})\bigr)+\pi (\sqrt{x})-1$ as given .  My question is, why does $\bigl\lfloor\phi(x,\lfloor\sqrt{x}\rfloor)+\sqrt{x}-1\bigr\rfloor$ also equal $\pi(x)?$
Suffixes indicating people and language of country vary by country:  _an: American, German, Italian, Belgian, Australian, Russian, Ukrainian, Korean,  Mexican, Brazilian, Chilean, Argentinean, Indonesian, Malaysian, Iranian, Egyptian, Somalian, Palestinian  _sh: English, British, Polish, Danish, Swedish, Turkish,    _ch: French, Dutch  _ese: Chinese, Taiwanese, Japanese, Vietnamese, Portuguese  _no: Filipino, Latino  _i: Israeli  _: Thai  Among the above, _an seems to outnumber other suffixes.  Are there any simple rule to govern appropriation of suffix turning a country name into adjective, indicating people and language of countries in the world in terms of say, region, race, or cultural background? Or is it just discretional? Get steam to realize I have games already installed I uninstalled and re-installed steam. I didn't uninstall my games, and they still exist on my hard drive (I checked).  I have an SSD and a HDD, and to save space I install programs (including steam) to C:\Program Files (x86)\Steam drive (SSD) and games to D:\Games\SteamApps (HDD). How to I let steam know the programs are installed there? Determining Min - Max within zip code (statistics by area)? I'm new to ArcGIS Desktop. I have created a layer representing zip codes and overlayed it with a layer containing some data points. I want to find the min and max value of the data points in each zip code. Currently, I am manually checking the values if all the data points in each zip code but this is becoming very time-consuming and cumbersome.   Is there a faster way to do this? Correlations between continuous and categorical (nominal) variables I would like to find the correlation between a continuous (dependent variable) and a categorical (nominal: gender, independent variable) variable. Continuous data is not normally distributed. Before, I had computed it using the Spearman's $\rho$. However, I have been told that it is not right.  While searching on the internet, I found that the boxplot can provide an idea about how much they are associated; however, I was looking for a quantified value such as Pearson's product moment coefficient or Spearman's $\rho$. Can you please help me on how to do this? Or, inform on which method would be appropriate?  Would Point Biserial Coefficient be the right option? How can I change the umask for one command only? How can I interactively execute a command in Linux (zsh, if it matters) with a different umask from the default, for one command only? Perhaps a combination of commands combined in a single line? The new umask should apply only to that command and return to its default value for the next command entered.  Ideally the command would be agnostic to the default umask in force before its entered (in other words, the default umask doesn't have to specified). What are the differences between a HashMap and a Hashtable in Java? What are the differences between a  and a  in Java?  Which is more efficient for non-threaded applications? Variable declaration placement in C Is losing privileges after placing bounty OK? My rep went over 2,000 today which meant I could edit other people's posts. Great! Then I added a bounty to a previous question, which brought me back down to 1,984...and I can no longer edit other posts. Should this changed? Or is this ideal? I'm looking to cite a number of web pages using bibtex and I was wondering if there was a specific template of the form @&lt;template name here&gt; for doing that. If you could use the following website as an example that would be great I think that the question is contained in the title. Suppose we begin from something that is false for example $1=0$. Is it possible using only $\Rightarrow$ (and of course $\lnot ,\wedge,\lor$)  to prove any possible statement? It has been ages since i studied logic at the university so i would prefer a simple explanation (if possible) According to the , you can harden a MySQL server by adding passwords, or removing the anonymous accounts.      If you want to prevent clients from connecting as anonymous users   without a password, you should either assign a password to each   anonymous account or else remove the accounts.   Before hardening, my users table looked like this.   mysql&gt; select user,host,password from mysql.user; +------------------+-----------+-------------------------------------------+ | user             | host      | password                                  | +------------------+-----------+-------------------------------------------+ | root             | localhost | *F3A2A51A9B0F2BE246XXXXXXXXXXXXXXXXXXXXXX | | root             | gitlab    |                                           | | root             | 127.0.0.1 |                                           | | root             | ::1       |                                           | |                  | localhost |                                           | |                  | gitlab    |                                           | | debian-sys-maint | localhost | *95C1BF709B26A5BAXXXXXXXXXXXXXXXXXXXXXXXX | | myuser           | localhost | *6C8989366EAF75BB6XXXXXXXXXXXXXXXXXXXXXXX | +------------------+-----------+-------------------------------------------+   I've remove all anonymous accounts, so that the user table now looks like this.  (I'm using puppet to manage the users, but puppet effectively performs a DROP USER command).   mysql&gt; select user,host,password from mysql.user; +------------------+-----------+-------------------------------------------+ | user             | host      | password                                  | +------------------+-----------+-------------------------------------------+ | root             | localhost | *F3A2A51A9B0F2BE246XXXXXXXXXXXXXXXXXXXXXX | | debian-sys-maint | localhost | *95C1BF709B26A5BAXXXXXXXXXXXXXXXXXXXXXXXX | | myuser           | localhost | *6C8989366EAF75BB6XXXXXXXXXXXXXXXXXXXXXXX | +------------------+-----------+-------------------------------------------+   Why is it that I am still able to login to my test system without a username or a password? What do I need to do to prevent any unwanted users from logging in?   root@gitlab:~# mysql Welcome to the MySQL monitor.  Commands end with ; or \g. Your MySQL connection id is 98 Server version: 5.5.32-0ubuntu0.12.04.1 (Ubuntu) .... mysql&gt;    Update:  I've also just discovered that I can login as root, without entering a password.  Update2: I found  which has some good information, but does not solve the issue.   There are no anonymous users.  mysql&gt; select user,host,password from mysql.user where user=''; Empty set (0.00 sec)   I login as root@localhost.  mysql&gt; select USER(),CURRENT_USER(); +----------------+----------------+ | USER()         | CURRENT_USER() | +----------------+----------------+ | root@localhost | root@localhost | +----------------+----------------+    I do not have a default password, or skip-grant-tables defined in my.cnf  root@gitlab:~# cat /etc/mysql/my.cnf |grep -i 'skip-grant-tables'|wc -l 0 root@gitlab:~# cat /etc/mysql/my.cnf |grep -i 'pass'|wc -l 0   Update3:  I have tried performing these steps with puppet, (which should perform a flush privileges automatically). I have also manually flushed privileges and also tried restarting mysql.   Update4: I've also tried changing the mysql root password and flushed privileges. No luck, I can still log in as any user without a password. How to search for email of a particular date from years ago in Gmail There are similar questions and several answers. I tried to follow the methods provided in the answers.  They did not work. So I am going to ask it again with more specifics.  I only knew the date of the email in the in-box in 2012 and the content of it but did not know the exact wording of the subject. Is it possible to find this email? What is a decoupling capacitor and how do I know if I need one? What is a decoupling capacitor (or smoothing capacitor as referred to in the link below)?  How do I know if I need one and if so, what size and where it needs to go?    mentions many chips needing one between VCC and GND; how do I know if a specific chip is one?   Would an  4-bit parallel access shift register used with an Arduino need one? (To use my current project as an example) Why or why not?  I feel like I'm starting to understand the basics of resistors and some places they're used, what values should be used in said places, etc, and I'd like to understand capacitors at the basic level as well. The layman's question. Is there a maximum temperature that can be reached by air on Earth due to global warming? For example, one that would prevent living organisms from functioning? I read somewhere that with the increase of the sun's brightness by 10%, the average temperature on our planet will be 47 C. This, however, only for about 600 million years. Would global warming enable reaching such a high average temperature? sum of noncentral Chi-square random variables I need to find the distribution of the random variable  $$Y=\sum_{i=1}^{n}(X_i)^2$$ where $X_i\sim{\cal{N}}(\mu_i,\sigma^2_i)$ and all $X_i$s are independent. I know that it is possible to first find the product of all moment generating functions for $X_i$s, and then transform back to obtain $Y$'s distribution. However, I wonder whether there is a general form for $Y$ like the Gaussian case: we know the sum of independent Gaussian is still a Gaussian, and thus we only need to know the summed mean and summed variance.   How about all $\sigma^2_i=\sigma^2$? Will this condition make a general solution? KOMA-Classes (scrbook, scrreprt, ... ): Disable chapter separation in \listoffigures and \listoftables I use \listoffigures and \listoftables. Both commands create line breaks for items from different chapters. How can you stop that?  Screenshot: Where to place 'only' relative to prepositions? I know that questions about the placement of 'only', are often asked here; accordingly, I searched for an answer to my question before posting it.  Question  Where are focusing adverbs placed relative to prepositions?  For example,   For his presentation, he needed to talk only about tomatoes. For his presentation, he needed to talk about only tomatoes.   My Attempt to Answer the Question  In both examples, it seems to me that 'only' is placed close enough to what it is intended to modify (the topic of the subject's discourse: tomatoes) that I cannot see any justifiable misinterpretations.  One of the examples sounds better to me, but at this point in my studies, I am more interested in what the traditional rules of grammar require, than in how the wordings sound. I want to shrink my second (LVM) partition, in order to create a new partition in the newly freed space. I am using the Live CD to do so, because I know I can't resize/move this partition while it is in use. When I opened GParted in the Live CD, I realized that I could not resize the partition, because when I right-click it, the option "resize/move" is disabled.  I tried to unmount it, to "lazy" unmount it (umount -l /dev/sda2) but it didn't work.  A screenshot from GParted: How to add Android Studio to the launcher? How to get a list of applications installed in Ubuntu dash? In Gnome, we have applications grouped up based on their type (like - games, accessories, et al). But in Unity, we need to manually type in the Unity to get the App.  Is there any keyword/utility to display the applications in GUI?  Ubuntu Version : 14.04 Hardware : Dell Inspiron 3437