ζ(3)

Definition

ζ(3) refers to the value of the Riemann zeta function evaluated at 3, which is approximately equal to 1.2020569. This special value has significant implications in number theory, particularly in relation to various sums and series. It is known to be irrational, and its connections to other mathematical constants and functions have intrigued mathematicians for centuries.

5 Must Know Facts For Your Next Test

1. ζ(3) is known as Apéry's constant because Roger Apéry proved its irrationality in 1979.
2. The evaluation of ζ(3) can be related to multiple series, such as the sum of the reciprocals of cubes: $$rac{1}{1^3} + rac{1}{2^3} + rac{1}{3^3} + ...$$
3. Unlike ζ(2), which is linked to $$rac{ ext{π}^2}{6}$$, the exact closed form of ζ(3) remains unknown.
4. ζ(3) appears in various areas of physics, including quantum field theory and statistical mechanics, demonstrating its broader significance beyond pure mathematics.
5. There are several conjectures relating to ζ(3), including its values at rational multiples of $$ ext{π}$$ and its connections to modular forms.

Review Questions

• How does the value of ζ(3) relate to the study of series and sums in number theory?
  • ζ(3) represents the sum of the reciprocals of cubes, which can be expressed as $$rac{1}{1^3} + rac{1}{2^3} + rac{1}{3^3} + ...$$ This connection highlights how special values like ζ(3) can arise from evaluating series, making it an important aspect of summation in number theory. Understanding this relationship helps to appreciate the properties and significance of ζ(3) within the broader framework of infinite series.
• Discuss the significance of Apéry's proof regarding the irrationality of ζ(3) and its impact on number theory.
  • Apéry's proof that ζ(3) is irrational marked a significant milestone in number theory. Prior to this proof, many mathematicians speculated about the nature of special values of the Riemann zeta function but had not conclusively established irrationality for odd integers greater than one. The result not only enhanced interest in ζ(3) but also led to further research on similar constants, deepening our understanding of their properties and implications within mathematical theory.
• Evaluate the ongoing research regarding the value and properties of ζ(3), particularly in connection with other mathematical concepts.
  • Current research on ζ(3) continues to explore its relationships with other mathematical constants and theories, particularly its connection to modular forms and possible closed forms involving rational multiples of $$ ext{π}$$. Investigations into its occurrences in physics have sparked interest as well, leading to potential new insights into quantum field theories. As mathematicians strive to unveil deeper connections among these values, the exploration surrounding ζ(3) is likely to yield important developments in both number theory and mathematical physics.

Related terms

Riemann Zeta Function: A complex function defined for complex numbers, which plays a central role in number theory and has applications in many areas of mathematics.

Bernoulli Numbers: A sequence of rational numbers that are deeply connected to the values of the Riemann zeta function at positive integers, particularly even integers.

Special Values: Specific outputs of mathematical functions that hold particular importance or unique properties, such as ζ(3) being connected to sums of powers of integers.