Here I am considering the original zeta function (not the extended one)$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$ When Re(s)>1, the Zeta function converges, and if Re(s)&lt;1 it diverges.  Here is my question. What happens when Re(s)=1?   I know it diverges when s=1, but does it converge otherwise?
Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge?
Find all solutions to the equation $(z+1)^6=z^6$. Express the solutions in simplified rectangular form. Hint: Clearly $z=0$ is not a solution. $$(z+1)^6=z^6 $$ $$\frac{(z+1)^6}{z^6 }=1$$ $$\left(\frac{(z+1)}{z }\right)^6=1$$ $$\left(\frac{(z+1)}{z }\right)^6=\cos(2*\pi*k)+i\sin(2*\pi*k)$$ $$\frac{(z+1)}{z }=\left(\cos(2*\pi*k)+i\sin(2*\pi*k)\right)^{1\over6}$$ $$1+\frac{1}{z }=\cos({2*\pi*k\over6})+i\sin({2*\pi*k\over6})$$  $$\frac{1}{z }=(\cos({\pi*k\over3})-1)+i\sin({\pi*k\over3})$$ $$\frac{1}{(\cos({\pi*k\over3})-1)+i\sin({\pi*k\over3}) }=z$$ I dont know how else to solve this. A lot of help is appreciated. Also i know that k should go from 1-5. How do I move child windows without moving or minimizing parent in GNOME 3? Google page title not matching html  tag Long wait time on login I am a Bangladeshi citizen. Back in 2013, I went to India with a tourist visa and overstayed for four days. It was due to a gout attack in my feet but I don't have my medical papers to prove that.  I applied again in 2015 for a visa, at Dhaka, and they refused my visa and blacklisted my passport, writing some numbers and date on it. Now I am not able to get an Indian visa.  How I can remove my passport from the blacklist and get a new Indian visa? Do I need a Indian immigration lawyer to do this? How can I select first occurrence between two patterns including them. Preferably using sed or awk.  I have:  text something P1 something content1 content2 something P2 something text something P1 something content3 content4 something P2 something text   I want the first occurrence of the lines between P1 and P2 (including P1 line and P2 line):  something P1 something content1 content2 something P2 something The proof I have starts of with $\;xy=0\;$ in a field.   Then $x^{-1}$ exists because it is a field.  Then $x^{-1} xy=x^{-1} 0$.  Therefore $y=0$.  But surely if an integral domain can not have any zero divisors, how can we end the proof by saying $y=0$? Surely then y is a zero divisor and hence the field is not an integral domain? sed fails with "unknown option to `s'" error What should I look for when buying chain lube / lubricant? In the past, I personally have only used WD-40.   However, I have occasionally had my bicycle looked over at the locally run, free service sessions and most of the mechanics there use a hand applied type of greasy gel (I forget the name).  Questions:   Is WD-40 suitable for bicycle chains? Would alternatives provide better chain longevity? When shopping around for lubricant, what are the most important things to look for? What do you use and why?  (optional added extra) Any tips for reducing chain wear? There is no set of all sets.. So there are a few posts about this already, but they skip over the problem I have.  The proof of Cantor's theorem elegantly shows that if we consider $f:A\rightarrow \mathcal{P}(A)$, the set $B=\{x\in A : x\not\in f(x) \}$ by its definition cannot lie in the image of $f$, precluding it from being a bijection - hence power sets have strictly greater cardinality.  But now, I thought one (obvious) way to characterise the "set of all sets" would be $S=\mathcal{P}(S)$, as indeed, it contains all possible sets (perhaps here is the problem, but I don't immediately see how using this as a definition for the set of all sets is bad). Now consider simply taking the identity function $i:S\rightarrow S$. Then $B=\{x\in S : x\not\in x \}$ - the set of all sets which do not contain themselves. But this doesn't exist. It's exactly the set of Russell's paradox. If this isn't a set, and so is not in $S$, then the proof no longer offers an obstruction to $i$ being surjective.    Each other post I've seen on the matter uses Cantor's theorem to show such a set doesn't exist. My query is to find that something else that must be happening, since I think the proof has issues for this particular set. I'm expecting other issues to arise with the definition $S=\mathcal{P}(S)$... How should an educational institute woo good faculty? Installing an addon There is this addon that was built called Window Generator 2. It can be . However, when I go to the download link it shows up in a new tab as a bunch of code.  How can I download and use this addon in Blender? Javascript get date in format Is there a video that explains how Stack Overflow works and can be used? Is there a video that explains how Stack Overflow works and can be used? I.e. a short tutorial to get new users started. If there isn't, then the people behind this site or others who know how it works should make one, because it's overwhelmingly different from other sites and thus difficult for beginners. How to change the hostname of a RHEL-based distro? 0 byte file created in a simple script I have a script as below:  #!/bin/bash df -k | tr -s " " "," | awk 'BEGIN {FS=","} {print $1,$5}'|sed 1d &gt; file1.txt while read partition percentUsed do if [ $percentUsed &gt; 75 ] then echo Partition: ${partition} space is ${percentUsed} else echo Pration: $partition: OK!! fi done &lt; file1.txt   The script is executing properly, however, it is creating a zero byte file by the name 75. Any ideas why this is happening?  $ sh diskUsed.sh Partition: C:/Users/Public/Documents/CYGWIN/bin space is 75% Partition: C:/Users/Public/Documents/CYGWIN/lib space is 75% Partition: C:/Users/Public/Documents/CYGWIN space is 75% Partition: C: space is 75% Partition: H: space is 91%   $ ls -lrt total 2 -rwxrwxrwx 1 diwvibh Domain Users 284 Nov 21 04:17 diskUsed.sh -rw-r--r-- 1 diwvibh Domain Users 133 Nov 21 04:29 file1.txt -rw-r--r-- 1 diwvibh Domain Users   0 Nov 21 04:29 75 How do I set up file sharing between two Ubuntu laptops on my wireless network? I have two laptops running Ubuntu 12.04. Both connect to the same WIFI network to access internet.  How do I connect them to each other so that I can access files on one from another and vice versa?  Also, how do I manage the user permissions, etc. on them? Stupidly, I had been using a condition like this as part of a script:  if [ $(ls FOO* 2&gt; /dev/null) ] # if files named "FOO*" were downloaded  then     echo "Files found"      # ... process and email results else     echo "Not found"     # ... email warning that no files were found (against expectations) fi   That works for zero and one files named FOO*, but fails if there are more than one. From logs I found several different error messages stemming from this:  [: FOO_20131107_082920: unary operator expected [: FOO_20131108_070203: binary operator expected [: too many arguments   My question is: what is the correct way to check, in a Bash if condition, whether one or more files whose name begins with FOO exist?  GNU bash, version 4.2.25(1)-release (x86_64-pc-linux-gnu) Prepend 0 to sublists Check if a class is derived from a generic class I have a generic class in my project with derived classes.  public class GenericClass&lt;T&gt; : GenericInterface&lt;T&gt; { }  public class Test : GenericClass&lt;SomeType&gt; { }   Is there any way to find out if a Type object is derived from GenericClass?  t.IsSubclassOf(typeof(GenericClass&lt;&gt;))   does not work.