Question

The series \(\sum_{n=1}^{\infty} 1 / n^{s}, s>1,\) is called the Riemann Zeta function, \(\zeta(s) .\) \((\text { a })\) you found \(\zeta(2)=\pi^{2} / 6 .\) When \(n\) is an even integer, these series can be summed exactly in terms of \(\pi .\) ) By computer or tables, find $$\text { (a) } \quad \zeta(4)=\sum_{n=1}^{\infty} \frac{1}{n^{4}}$$ $$\text { (b) } \quad \zeta(3)=\sum_{n=1}^{\infty} \frac{1}{n^{3}}$$ $$\text { (c) } \quad \zeta\left(\frac{3}{2}\right)=\sum_{n=1}^{\infty} \frac{1}{n^{3 / 2}}$$

Short answer

\(\zeta(4) = \frac{\pi^4}{90}\), \(\zeta(3) \approx 1.2020569\), \(\zeta(3/2) \approx 2.6123753\)

Step-by-step solution

Definition and known values

The Riemann Zeta function is defined as \[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \quad \text{for} \quad s > 1 \] It is given that \( \zeta(2) = \frac{\pi^2}{6} \). We will use similar methods and known values to find \(\zeta(4)\), \(\zeta(3)\), and \(\zeta(3/2)\).

Find \(\zeta(4)\)

Using known tables or mathematical software, \[ \zeta(4) = \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90} \] This value is derived from more advanced techniques involving Fourier series or integral representations.

Find \(\zeta(3)\)

Unfortunately, \(\zeta(3)\) (often referred to as Apéry's constant) does not have a known representation in terms of \(\pi\). By numerical approximation, \[ \zeta(3) \approx 1.202056903159594 \] This value can be found using tables or computer software.

Find \(\zeta(3/2)\)

Similar to \(\zeta(3)\), \(\zeta(3/2)\) does not have a simple expression in terms of \(\pi\). By numerical approximation, \[ \zeta\left(\frac{3}{2}\right) \approx 2.612375348685488 \] This value can also be obtained using numerical methods or software.

Key concepts

convergence of series

The Riemann Zeta function is defined as the infinite series \(\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}\) for \(s > 1\). To understand why this series converges, we need to look at the behavior of the terms as \(n \to \infty\). As \(n\) gets larger, each term \(\frac{1}{n^s}\) becomes smaller. This means that the series will eventually sum to a finite number, provided \(s > 1\). For example, in the original exercise, finding \(\zeta(2) = \frac{\pi^2}{6}\) demonstrates that the series converges to a specific value for \(s = 2\).
\Examples of series convergence can also be seen in other mathematical functions. It is essential for the computation and assessment of more complex sums.
There are also other criteria to test the convergence of series like the Comparison Test, the Integral Test, and the Ratio Test that further solidify our understanding.

special functions in mathematics

The Riemann Zeta function is a fundamental example of a special function in mathematics. Special functions often arise in mathematical physics, number theory, and other applied fields. In the context of the Zeta function, one of its special properties is its relationship with prime numbers, encapsulated by the Euler product formula: \[\zeta(s) = \prod_{p \text{ prime}} \left(1 - \frac{1}{p^s} \right)^{-1} \] for \(s > 1\).
The function also appears in many important mathematical theorems, like the Riemann Hypothesis, which conjectures that all non-trivial zeros of the Zeta function have real part \(\frac{1}{2}\).A special function, like the Zeta function, often has unique and intricate properties that make them useful for solving problems.
Understanding the Zeta function helps in grasping these deeper mathematical relationships.

numerical approximations

Not all values of the Riemann Zeta function can be expressed in simple closed forms. For instance, \(\zeta(3)\), known as Apéry's constant, is not expressible in terms of elementary functions. We thus rely on numerical approximations. Using mathematical software or tables, we find \[\zeta(3) \approx 1.202056903159594 \].
Similar methods are used for other values, such as \[\zeta\left(\frac{3}{2}\right) \approx 2.612375348685488 \].Numerical approximations are essential for practical computations when exact values are unknown. Efficient algorithms and high-precision arithmetic allow us to get very accurate values.
These approximations give mathematicians and scientists the tools to use these functions in real-world problems, where exact values might be hard to come by.