"Why" is the Euler-Mascheroni constant near sqrt(1/3)?

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skullturf
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"Why" is the Euler-Mascheroni constant near sqrt(1/3)?

Postby skullturf » Wed Feb 05, 2014 10:22 pm UTC

The Euler-Mascheroni constant (also called "Euler's constant", or lower case gamma) is approximately equal to 0.57722.

The square root of 1/3 is approximately equal to 0.57735.

I remember noticing this a while ago, and wondering if there was any informal or intuitive "reason" for it to be true.

Obviously this question is highly subjective, and in a sense, many of these types of things are merely coincidences.

However, there are some arguments out there that do a pretty good job of being intuitive or informal reasons for some of these coincidences. For instance, the coincidence that Pi^2 is slightly less than 10, or the coincidence that Pi is slightly less than 22/7.

If we take the result Sum(1/n^2) = Pi^2/6 as "known", there's a cute argument due to Noam Elkies that compares Sum(1/n^2) to a telescoping series and shows that Pi^2/6 is slightly less than 10/6.

And there's a proof that 22/7-Pi is a small positive number, which consists of evaluating a definite integral of a certain "small" nonnegative rational function.

I asked about this on math.stackexchange, but a lot of questions get asked there, and sometimes questions get lost in the shuffle if not answered quickly.

In fact, someone left a comment linking to this, which looked promising at first glance, but I had trouble filling in the details.

So, if we express the Euler-Mascheroni constant as an integral, can anyone help me come up with a way of approximating that integral which shows that its value must be "close" to the square root of 1/3?

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Re: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?

Postby mike-l » Thu Feb 06, 2014 6:23 am UTC

Mostly coincidence as far as I can tell. Especially the pi ones, as 10 and 22/7 aren't particularly interesting numbers (10 only seems interesting because we happen to have 10 fingers and so picked it as our base). And pi^2 is relatively far from 10, 0.13 off. 26% of numbers are within 0.13 of some integer.

The 22/7 is slightly more interesting, as it's a better approximation than you would expect with such a small denominator. Indeed, it's in some way the best approximation with such a small denominator. See, continued fractions. This also relates to the Euler-Mascheroni constant and 1/root(3), as their continued fractions agree for 7 terms, so they are good approximations for each other. What's actually interesting here is that the constant has a nice small continued fraction for the first 7 terms, which is why it's comparable to other 'nice' numbers.

Specifically, both continued fractions start with [0;1,1,2,1,2,1] = 15/26, and so both must be nearer to 15/26 than to any fraction with denominator less than 26. In fact they are both within 1/2340th of 15/26.
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Re: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?

Postby Bloopy » Fri Feb 07, 2014 2:50 pm UTC

skullturf wrote:In fact, someone left a comment linking to this, which looked promising at first glance, but I had trouble filling in the details.

I'll guess coincidence for this part for a start. The 2nd-degree Legendre polynomial has roots ±sqrt(1/3) because it's an integral of 3x such that f(1) = 1, and the math works out nicely for its uses. Approximating a value close to sqrt(1/3) using a function involving sqrt(1/3) and then claiming there's a link seems like circular reasoning to me. I imagine you'd get the same result with many other functions that have integral from 0 to 1 close (or equal) to gamma.

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Re: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?

Postby skullturf » Tue Feb 11, 2014 5:03 pm UTC

That's very similar to what I thought at first. If the number sqrt(1/3) comes from the method of approximate integration, as opposed to the function being integrated, then it does indeed seem like a bit of a cheat.

However, somewhat ironically, when I tried to work through the details of that Wikipedia comment (i.e. applying two-point Gauss-Legendre quadrature to the integral of 1/x), I found that when using the particular function 1/x, the sqrt(1/3) disappeared! I got *some* sort of approximation for gamma, but it didn't contain sqrt(1/3) at all!

Some more details are in the attached PDF.

Maybe I missed something when I tried to fill in the details of the Wikipedia comment. Is there another way to interpret the suggestions in the comment that does end up giving sqrt(1/3) as one's approximation to gamma?

I do admit that ultimately, these kinds of things are probably just coincidences. There simply might not be an "intuitive" answer for "why" sqrt(1/3) is about 0.57735 and gamma is about 0.57722.

However, what if we relax the question a little bit? Can anybody help me find "good" ways of approximating gamma, involving clever use of integrals, series, etc., without worrying about sqrt(1/3) as our "target"? If you lived in the pre-computer age and wanted four or five digits of gamma, what specific manipulations would you do?
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Re: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?

Postby Dthen » Tue Feb 11, 2014 9:50 pm UTC

mike-l wrote:The 22/7 is slightly more interesting, as it's a better approximation than you would expect with such a small denominator. Indeed, it's in some way the best approximation with such a small denominator.


I've never understood why people use 22/7 as an approximation of Pi. 22/7 is four characters, but is only accurate to 3.14, which is also 4 characters ... you might as well just remember 3.14. A better approximation is 355⁄113.
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Re: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?

Postby Flumble » Tue Feb 11, 2014 11:22 pm UTC

Dthen wrote:I've never understood why people use 22/7 as an approximation of Pi. 22/7 is four characters, but is only accurate to 3.14, which is also 4 characters ...

A) because it's a slightly better approximation: |log(pi)-log(22/7)| < |log(pi)-log(3.14)| (about 20% better on a logarithmic scale, however you want to interpret that :P )
B) because it has a small denominator, so it's useful when working with rationals —when doing calculations with other decimal numbers, you're most often better off with 3.14.

[edit] I accidentally a letter
Last edited by Flumble on Wed Feb 12, 2014 10:05 am UTC, edited 2 times in total.

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Dthen
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Re: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?

Postby Dthen » Wed Feb 12, 2014 8:22 am UTC

Hmm, I see. I suppose that sort of makes sense. Thanks.
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Re: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?

Postby quarkcosh1 » Thu Feb 13, 2014 9:31 pm UTC

The 22/7 thing also comes from a continued fraction representation of pi.

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Re: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?

Postby Bloopy » Mon Apr 14, 2014 10:53 pm UTC

It probably doesn't help, but I noticed these two integrals are a good approximation of each other geometrically, not just numerically:

Image
Image

http://www.wolframalpha.com/input/?i=pl ... +0%3Cx%3C1

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Re: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?

Postby cyanyoshi » Wed Apr 16, 2014 5:26 am UTC

That's a neat observation. It's also cool how Wolfram Alpha uses the digamma function to generalize the harmonic numbers.

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Re: "Why" is the Euler-Mascheroni constant near sqrt(1/3)?

Postby skullturf » Wed Apr 16, 2014 10:04 pm UTC

Thanks for that, Bloopy.

That's a good example of the type of thing I was missing, and hoping for.

I knew how to write gamma as a slowly converging series, but I didn't know about generalizing the harmonic numbers to non-integer arguments, nor did I know about that specific definite integral that evaluates to gamma.

EDITED TO ADD: For anyone reading this who is curious, a bit more information can be found here:

http://en.wikipedia.org/wiki/Harmonic_n ... _arguments


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