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Euler’s Constants

September 5, 2013


Having a constant named after you is cool

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Jeffrey Lagarias was a Distinguished Member of Technical Staff at AT&T Bell Laboratories for years, and later joined the math department at the University of Michigan, while still keeping involved with the “labs.” His early background is in deep aspects of number theory, and he still works in that area. He has also broaden his interests into other parts of mathematics and theoretical computer science.

Today I wish to talk about a beautiful survey paper he has just published on Euler’s constant in the AMS Bulletin.

Leonhard Euler has at least two constants named for him: the base of the natural logarithms, which is, of course, the constant {e=2.718\dots}, and the more mysterious constant {\gamma}. It is pretty cool to have a constant named after you. Many have theorems named for them, others have methods—few have constants. See our friends at Wikipedia here for some other examples:

  • Archimedes’ constant is {\pi}. But it is not actually named for him, so perhaps it does not really belong here.
  • Brun’s constant {B_{2}} is named for Viggo Brun, who proved that the sum

    \displaystyle  B_{2} = \sum_{p} \frac{1}{p}

    over the twin primes is finite. An estimate is that {B_{2}} is about

    \displaystyle  1.902160583104\dots

    Yitang Zhang’s recent breakthrough—see here—shows that this number is even more important.

  • Roger Apéry shocked everyone when he proved that {\zeta(3)} is not rational. His constant is therefore the value of the famous zeta function at {3}, which is approximately

    \displaystyle 1.202056903159594285399738161511449990764986292\dots

    Just approximately.

Euler’s Constant

Euler’s {\gamma} is defined by

\displaystyle  \gamma = \lim_{n \rightarrow \infty} \left( \sum_{k=1}^{n} \frac{1}{k}- \log n \right).

It is not obvious that this is even well-defined, but it is:

\displaystyle  \gamma = 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 \dots

Alexander Yee and Raymond Chan have computed it to a few more places, {29,844,489,545} more digits to be exact.

The main question about the Euler constant {\gamma} is: Is it a rational number? This is one of those very difficult problems that seem beyond the abilities of mathematics as we know it. But as with {\zeta(3)}, perhaps someone with an inspiration like Apéry’s will come along and prove that it is not rational. You never know until you know.

Euler’s Research Rules

In Jeff’s wonderful paper there are many extremely interesting results and facts about the constant {\gamma}, and we invite you to study them. But here I’d like to highlight his formulation and summary of rules that Euler essentially followed:

  1. Always attack a special problem. If possible solve the special problem in a way that leads to a general method.
  2. Read and digest every earlier attempts at a theory of the phenomenon in question.
  3. Let a key problem solved be a father to a key problem posed. The new problem finds its place on the structure provided by the solution of the old; its solution in turn will provide further structure.
  4. If two special problems solved seem cognate to each other, try to unite them in a general scheme. To do so, set aside the differences, and try to build a structure on the common features.
  5. Never rest content with an imperfect or incomplete argument. If you cannot complete and perfect it yourself, lay bare its flaws for others to see.
  6. Never abandon a problem you have solved. There are always better ways. Keep searching for them, for they lead to a fuller understanding. While broadening, deepen and simplify.

Open Problems

I should remember to follow these rules myself. As for the constant {\gamma} I have great interest in its status—rational or not?—and no idea at all of how to proceed. Perhaps someone will soon.

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6 Comments leave one →
  1. September 5, 2013 8:55 am

    Benjamin Peirce liked Euler’s Formula e^{i\pi} + 1= 0 so much that he introduced three special symbols for e, i, \pi — ones that enable e^{i\pi} to be written in a single cursive ligature, as shown in this note.

  2. dreamthorn permalink
    September 6, 2013 2:25 am

    I thought I found a ridiculously simple proof that Euler’s gamma is irrational, but then I noticed that it is possible to add two irrational numbers to get a rational number…

    It seems you have managed to inspire me to investigate this problem :)

  3. o0o permalink
    September 6, 2013 5:02 pm

    Lagarias’ “elementary” problem equivalent to the Riemann hypothesis is beautiful:
    \sigma(n) \le H_n + e^{H_n} \ln H_n \,

  4. Peter L. Griffiths permalink
    November 29, 2013 1:42 pm

    Historically Euler’s constant originates from the Supplement to De Moivre’s Miscellanea Analytica (1730). However De Moivre expresses the difference between the Harmonic series and the log of the next fraction as an expression containing the Bernoulli numbers which unfortunately diverge. Euler corrects this divergence in his E47 paper (1735) page 118 which also contains his value of the constant.

Trackbacks

  1. Euler’s Constants | Rocketboom
  2. Euler’s Research Rules | Dana C. Ernst

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