Can anyone please explain to me how to prove that Riemann zeta function is 0 for all negative even numbers. In many references , they have just given the statement without any proof. Any explanation or hint will be a great help. Thanks in advance.
The Riemann zeta function $ζ(s)$ is defined for all complex numbers $s ≠ 1$ with a simple pole at $s = 1$. It has zeros at the negative even integers, i.e., at $s = −2, −4, −6, ...$.   My question: How one can obtain these roots.
copy and paste doesn't work in terminal font-not-found error: font "HoeflerText" cannot be found -- LuaLaTex error! I am running this CV/Resume example from :   Here is a relevant fraction of the code:  \documentclass[10pt]{article} %!TEX encoding =  UTF-16 % See geometry.pdf to learn the layout options. There are lots. \usepackage[hmargin=1.25cm, vmargin=1.5cm]{geometry} %symbols - the ones you see on the left of the email and of the phone \usepackage{bbding} \usepackage{amssymb} \usepackage{eurosym} %Colors/Graphics \usepackage{color,graphicx} \usepackage[usenames,dvipsnames]{xcolor}  %Fonts and Tweaks for XeLaTeX \usepackage{fontspec,xltxtra,xunicode} \defaultfontfeatures{Mapping=tex-text} \setromanfont[Mapping=tex-text]{Hoefler Text} ...   So... when compiling with LuaLaTeX (part of MiKTeX on Windwos XP x64), I am getting the following error:  ... luaotfload | Font names database loaded: C:/Documents and Settings/Ampleforth/L ocal Settings/Application Data/MiKTeX/2.9/luatex-cache/generic/names/otfl-names .lua luaotfload | Updating the font names database: luaotfload | Scanning TEXMF and OS fonts... luaotfload | Font names database saved: %s  : C:/Documents and Settings/Ampleforth/Local Settings/Application Data/MiKTeX/2 .9/luatex-cache/generic/names/otfl-names.lua  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !  ! fontspec error: "font-not-found" !  ! The font "HoeflerText" cannot be found. !  ! See the fontspec documentation for further information. ! For immediate help type H &lt;return&gt;. !...............................................    l.28 \setromanfont[Mapping=tex-text]{Hoefler Text}   What is wrong? How can I install this font so LuaLaTeX is picking it up? Where to download? Any other thoughts about this? Why should edits be at least six non-space characters? Earlier today I got this message:  Oops! Your edit couldn't be submitted because:   Edits must be at least 6 non-space characters; is there something else to improve in this post?   However, the purpose of my edit was to change the formatting for a message to show code.  There may have been other grammatical mistakes that could have been, but in general it is possible that there would not be.  What's the correct protocol to handle a situation like this? Gilbert, MacWilliams, and Sloane defined a solution to "perfect integrity" in , unconditionally secure message authentication, not vulnerable to any cryptanalysis on its own even by an adversary with unbounded computational power, provided you have a key chosen uniformly at random the same length as the message. It sounds counter-intuitive that a MAC could have perfect integrity, any hash function can be broken with enough computational power. How does their solution have "perfect integrity"? I am travelling to Calgary from Delhi. I have an Air India flight from Delhi to Dubai then British Airways flight from Dubai to London and Air Canada flight from London to Calgary. My query is "Do I require a transit visa." Plz help as I am travelling in four days Difference between int[] array and int array[] I have recently been thinking about the difference between the two ways of defining an array:   int[] array int array[]   Is there a difference? evaluate $\int_{0}^{\infty}\cos(t) t^{z-1}dt=\Gamma(z)\cos(\frac{\pi z}{2})$ Alternative set theories This is a (soft!) question for students of set theory and their teachers.   OK: ZFC is the canonical set theory we all know and love. But what other, alternative set theories, should a serious student encounter (at least to the extent of knowing that the theory exists, and about why it is thought interesting)?  Suggestions might be SP (Scott-Potter), NBG, NF, ZFA, IST, ETCS (for a few sentences of explanation, see   But what would be on your list of alternative set theories would be it good for advanced students to hear about, if only briefly? QT MySql connectivity using Windows XP, Qt Creator 4.5.2(windows 32 bit) How to show or hide Keyboard Viewer with a keyboard shortcut? I would like to display and hide the Keyboard Viewer using a shortcut.   Is there a way to display the Keyboard Viewer via a keyboard shortcut? Is there a way to automatically display the Keyboard Viewer once I switch to a specific language? Is $L^{p}$ space with alternate norm Banach? Why are continuous functions the "right" morphisms between topological spaces? Is there a kind of microdata or markup that creates "results x of y" in search results I viewed in this post  a solution to my question, but...    Are there any easier way (or alternative) to do this using colors?  Thank you. ------------------ edited after showed than it´s duplicate ------------- I think that there are many sols, but for example this is easy using a normal table without pgf...     I´m trying to adapt to similar colors to apply to the table probabilitiess Let $X$ be a set and consider the collection $\mathcal{A}(X)$ of countable or cocountable subsets of $X$, that is, $E \in \mathcal{A}(X)$ if $E$ is countable or $X-E$ is countable. If $X$ is countable, then $\mathcal{A}(X)$ coincides with the power set $\mathcal{P}(X)$ of $X$. Now suppose that $X$ is uncountable. Assuming the axiom of choice, we can conclude that $\mathcal{A}(X) \ne \mathcal{P}(X)$, since $|X| = |X| + |X|$. So the question is:     Can we prove in ZF that $\mathcal{A}(X) \ne \mathcal{P}(X)$ for every uncountable set $X$?   I'm assuming that a set $X$ is uncountable if there is no injective function $f : X \rightarrow \mathbb{N}$. What is the relationship between redshift and conformal time ?   For example in a paper i found:   taking $z_e = 3234 $ at the time of radiaton-matter equality yields the conformal time $\frac{\eta_e}{\eta_0} = 0.007$ and taking $z_E=0.39$ at matter-$\Lambda$ equality yields $\frac{\eta_E}{\eta_0}=0.894$ and setting redshift at decoupling $z_d=1089$ yields $\frac{\eta_d}{\eta_0} = 0.0195$  where $\eta_0$ is the present decoupling time.  Further some cosmological parameters are given as :  $\Omega_r = 8.36 \times 10^{-5}, \Omega_m = \Omega_b + \Omega_{dm} = 0.044 +0.226, \Omega_\Lambda = 0.73,  H_0=0.72$  Now how can i calculate all those $\eta_e, \eta_E, \eta_d, \eta_0$ from given redshift values and/or above parameters ? I searched whole of my text books trying to find an explicit relation for conformal time $\eta$ but all i got was $ \mathrm{d}t=a(t)\mathrm{d}\eta$. Any help would be very helpful. Nikkor f/1.8G 35mm or 50mm? I'm new to photography. I got a Nikon D3100 with the 18-55 kit lens.  I was looking to buy a f/1.8 prime lens, and I found two options:   AF-S DX NIKKOR 35mm f/1.8G   AF-S DX NIKKOR 50mm f/1.8G   How should I go for choosing one or the other? How do I remove XFCE from my computer? My computer is overheating because I got too many desktop environments installed like: MATE, Cinnamon and XFCE.  I uninstalled them, but XFCE failed to uninstall. I get this error.   sudo apt-get purge xfce4     Reading package lists... Error!     E: Problem syncing the file - sync (5: Input/output error)     E: The package lists or status file could not be parsed or opened. How to open a PDF file from terminal? I used to work on the Mac terminal before and I used:   open file2open.pdf   and the PDF file would be opened on preview or whatever my default viewer was. When I use it in the terminal in Ubuntu I get this error message:   Couldn't get a file descriptor referring to the console CSS to make HTML page footer stay at bottom of the page with a minimum height, but not overlap the page