- #1
murshid_islam
- 457
- 19
i know that zeta(2) = pi^2/6 and zeta(4) = pi^4/90
what is zeta(3)? can i use Fourier series?
what is zeta(3)? can i use Fourier series?
Zeta(2) and Zeta(4) are special values of the Riemann zeta function, which is an important mathematical function used in number theory and analysis. Zeta(2) is equal to pi squared divided by 6, and Zeta(4) is equal to pi to the fourth power divided by 90. These values have been studied for centuries and have many interesting connections to other areas of mathematics.
The values of Zeta(2) and Zeta(4) were first discovered by the Swiss mathematician Leonhard Euler in the 18th century. He was able to derive these values using a technique known as analytic continuation, which extends the domain of a function beyond its original definition.
Zeta(3) is a transcendental number, meaning it cannot be expressed as a finite algebraic expression. Its value is approximately 1.2020569, but it is not known if there is a simpler or exact representation for this number. It is an important value in number theory, and has connections to other mathematical constants such as pi and e.
There is no known direct relationship between Zeta(3) and Zeta(2) or Zeta(4). However, all three values are special cases of the Riemann zeta function and are connected through various mathematical relationships. For example, Zeta(3) is related to the value of the Dirichlet eta function at 2, which in turn is related to Zeta(2).
Like Zeta(2) and Zeta(4), Zeta(3) has many important connections and applications in mathematics. It appears in various mathematical formulas and series, and has been studied extensively by mathematicians. It also has connections to other areas such as physics and engineering. However, the exact significance of Zeta(3) is still an active area of research in mathematics.