function

MATHS BITE: Apéry’s Constant

Apéry’s constant is defined as the number

{\displaystyle {\begin{aligned}\zeta (3)&=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}\\&=\lim _{n\to \infty }\left({\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+\cdots +{\frac {1}{n^{3}}}\right)\end{aligned}}}

where ζ is the Riemann Zeta Function.

This constant is named after the French mathematician Roger Apéry who proved that it was irrational in 1978. However it is still unknown whether or not it is transcendental.

History

The Basel Problem asked about the convergence of the following sum:
Screen Shot 2017-06-10 at 2.16.05 PM.png

In the 18th century, Leonhard Euler proved that in fact it did – to π^2/6. However, the limit of the following sum remained unknown:Screen Shot 2017-06-10 at 2.19.28 PM.png

Although mathematicians made some progress, including Euler who calculated the first 16 decimal digits of the sum, it was not known whether the number was rational or irrational, until Apéry.

Furthermore, it is currently not known specifically whether any other particular ζ(n), for n odd, is irrational. “The best we’ve got is from Wadim Zudilin, in 2001, who showed that at least one of ζ(5), ζ(7), ζ(9), ζ(11) must be irrational, and Tanguy Rivoal, in 2000, who showed that infinitely many of the ζ(2k+1) must be irrational.”

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