I recently noticed that these two numbers are remarkably close:  $$\frac{1}{\sqrt{3}}-\gamma=0.000135\dots$$     Are there any integrals or series which can prove that $\frac{1}{\sqrt{3}}&gt;\gamma$?   Meaning that (as usual in such cases) the function under the integral has to be non-negative and the value should be proportional to the difference of these numbers.  The same goes for the series (strictly non-negative terms).  A good overview for the inequalities with $\pi$, like $\frac{22}{7}&gt;\pi$, can be found in .    A . But I don't consider my question a duplicate, because the linked question is more general.
In 1735 Euler gave the value of $\gamma$ as $0.577218.$ The constant is generally defined as the limit of the difference between the harmonic series and $\log n:~\gamma= \lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{k}-\log n.$ Euler apparently relished this sort of calculation and must have taken quite a few terms to get such a good approximation.  My question is whether without a computer one can now prove that  $$\gamma &lt;  \sqrt{1/3}$$   with at least some savings in terms of the type of work Euler apparently expended? I don't think there's any point in raising the bar to $\gamma &lt; \ln 2\sqrt{\frac{23}{29}} $  because it seems to be the same sort of question.       My own thought was to compare $\frac{1}{2}\int_0^{1/3}\frac{dx}{\sqrt{x}}$ to something like $(-1)\cdot\int_0^\infty e^{-u}\ln u~du$ but I expect there's a better way.
What happened to the voting buttons? Starting from yesterday, the up-/downvote buttons seem to have moved from the left of a post towards above (in the middle) of the bodies of all questions and answers. Here is a screenprint of a sample (containing such buttons for both a question and an answer to it):    This happens on an iOS device (running 9.3.5), using various browsers and this in any SE site. No idea so far how it looks like in the most recent iOS version. Same for the (green) checkbox below those voting buttons (to mark an answer as accepted). Is this intended to be like this (IMO it looks ugly ... and a huge waste of available space), or is it rather some sort of CSS related issue that needs to be fixed? PS: Looking at the same page around the same time on a Win PC (using any of the common browsers) seems to work fine. What does the number in parentheses shown after Unix command names in manpages mean? I don't quite understand the difference between /a/ and /æ/. 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The full datasheet is available Is there to make the login screen appear on the external display in 18.04? On Ubuntu 18.04, I have connected my laptop to an external display and I am able to set the external display as primary and external display only mode. But when I turn on my laptop or log out from my session the display login screen is displayed only on my(internal) laptop screen. But once I login the display switches to external only and my laptop screen turns off as expected. Is there a way I can make the login screen appear only on the external screen on bootup or on logout as in 16.04? Also the login screen does not follow my cursor as in 16.04. How can I convert byte size into a human-readable format in Java? In many C/C++ macros I'm seeing the code of the macro wrapped in what seems like a meaningless do while loop.  Here are examples.  #define FOO(X) do { f(X); g(X); } while (0) #define FOO(X) if (1) { f(X); g(X); } else   I can't see what the do while is doing.  Why not just write this without it?  #define FOO(X) f(X); g(X)