Mirrors, Means, and Logarithms

Image from Dalhousie University

We’re pleased to announce the first math talk of 2010.  Dr. Dragan Jankovic of Cameron University will be talking this

Wednesday, February 10th at 5 PM in PHSC 1105.

His title and abstract are:

Title: Mirrors, Means, and Logarithms

Abstract. I will talk about an intrinsic symmetry of the logarithmic curve involving the classical means(arithmetic, geometric, and harmonic). In this context two curves naturally arise: one old and famous (yielding pi and many other things), and the other new (to me) and interesting (yielding Apery’s constant). In other words: a little bit of geometry, calculus, and history.

To intrigue you just a bit, let’s talk for a minute about Apery’s constant.  As we discussed here (and you learned in Calc III), the series

\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.

You surely also learned that the series

\sum_{n=1}^{\infty} \frac{1}{n^3}

also converges because it is a p-series with p=3.  The next question is: Which number does this series converge to?  The answer is Apery’s constant, \zeta_3, which is

\zeta_3 = 1.202056903159594285399738161511449990764986292...

Dr. Jankovic will no doubt have lots of interesting things to tell us, but we’ll mention two cool facts about \zeta_3:

  1. In 1978 Apery proved that \zeta_3 is irrational.  So, like \pi, the above decimal goes on forever.  Nobody knows if it is transcendental (ie. a number like \pi which is not the zero of any polynomial with integer coefficents).
  2. Apery’s constant also has close connections to number theory.  For example, amazingly, if you pick three random natural numbers, the probability that the three are relatively prime (ie. no natural number bigger than 1 divides all three) is exactly \frac{1}{\zeta_3} \sim .83.

P.S. To learn more about Apery’s constant and all your favorite other mathematical constants, you need to check out this webpage at Dalhousie University.  It is nothing less than a no-holds-barred cage match between mathematical constants!  Will Pi piledrive e?  Will Apery’s Constant put The First Surd in a headlock?  Check it out!

Math is taking all challengers who dare enter the Cage!

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