The Riemann Zeta Function and the Apéry's Constant: An Advanced Exploration

The Riemann Zeta Function, denoted as ζ(s), is a fundamental function in mathematics with wide-ranging applications in number theory, analysis, and physics. It is given by the analytic continuation of the series:

ζ(s) ∑n1∞ 1/ns

What is ζ(3)?

One of the most intriguing values of the Riemann Zeta Function is ζ(3), also known as the Apéry's constant. While it can be defined as:

ζ(3) ∑n1∞ 1/n3,

finding the exact sum of this series was a significant achievement in mathematics. Professor Roger Apéry provided a proof in 1978 that this constant is irrational, a fact that was previously unproven. The value of ζ(3) is approximately:

ζ(3) ≈ 1.202056903159594

Alternating Series Representation

Apéry provided a novel approach to represent ζ(3) using an alternating series. This representation is given by:

ζ(3) frac{5}{2} ∑n1∞ frac{(-1)n-1}{n3 binom{2n}{n}}

This form of the series has rapidly growing denominators, making it a powerful tool in understanding the behavior of the sum.

Additional Representations

There are numerous other ways to represent ζ(3), each with its unique properties and applications. One such representation is the generalized continued fraction due to Nesterenko:

ζ(3) frac{6}{5 - frac{1}{117 - frac{64}{535 - cdots - frac{n^6}{34n^3 - 51n^2 - 27n - 5}}}}

This continued fraction provides an alternative method to compute the value of ζ(3).

General Form of ζ(z)

The general form of the Riemann Zeta Function for any complex number ( z ) with Re(z) 1 is:

ζ(z) ∑n1∞ 1/nz

For the specific case of z 3, we get:

ζ(3) ∑n1∞ 1/n3 1 - 1/8 1/27 - 1/64 ... ≈ 1.202056903

This series is often approximated to 9 decimal places as:

Zeta(3) ≈ 1.202056903

Further Reading

To gain a deeper understanding of the Riemann Zeta Function and its values, you can visit:

The Riemann Zeta Function - Wikipedia Apéry's Constant - Wikipedia

These resources provide comprehensive information on the topic, making them invaluable for mathematicians, researchers, and enthusiasts alike.