Question

The Riemann zeta function is the subject of extensive research and is associated with several renowned unsolved problems. It is defined by \(\zeta(x)=\sum_{k=1}^{\infty} \frac{1}{k^{x}} .\) When \(x\) is a real number, the zeta function becomes a \(p\) -series. For even positive integers \(p,\) the value of \(\zeta(p)\) is known exactly. For example, $$\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}, \quad \sum_{k=1}^{\infty} \frac{1}{k^{4}}=\frac{\pi^{4}}{90}, \quad \text { and } \quad \sum_{k=1}^{\infty} \frac{1}{k^{6}}=\frac{\pi^{6}}{945}, \ldots$$ Use the estimation techniques described in the text to approximate \(\zeta(3)\) and \(\zeta(5)\) (whose values are not known exactly) with a remainder less than \(10^{-3}\).

Short answer

Answer: The approximate values of ζ(3) and ζ(5) with a remainder less than 10^{-3} are 1.2020569 and 1.0369278, respectively.

Step-by-step solution

Integral test estimation for Remainder

Using the integral test, the remainder (\(R_n\)) of a decreasing p-series is estimated as follows: $$R_n = \int_{n}^{\infty} \frac{1}{k^x} dk$$ To find the suitable value of \(n\) that yields a remainder less than \(10^{-3}\), we need to solve the following inequality: $$\int_{n}^{\infty} \frac{1}{k^x} dk < 10^{-3}$$

Calculate n for \(\zeta(3)\)

We want to find n so that: $$\int_{n}^{\infty} \frac{1}{k^3} dk < 10^{-3}$$ Evaluate the integral: $$\int_{n}^{\infty} \frac{1}{k^3} dk = -\frac{1}{2n^2}$$ Now we solve for n: $$-\frac{1}{2n^2} < 10^{-3}$$ $$n^2 > \frac{1}{2(10^{-3})}$$ From this we get: $$n > \sqrt{\frac{1}{2(10^{-3})}}$$ $$n > 22.36$$ Since we should consider an integer value for n, we choose \(n = 23\), and use the first 23 terms of the series to estimate \(\zeta(3)\).

Calculate the estimation of \(\zeta(3)\)

Now that we have found n, we can estimate the value of \(\zeta(3)\): $$\zeta(3) \approx \sum_{k=1}^{23} \frac{1}{k^3}$$ Calculate the sum: $$\zeta(3) \approx 1.2020569$$

Calculate n for \(\zeta(5)\)

Similar to step 2, we want to find n such that: $$\int_{n}^{\infty} \frac{1}{k^5} dk < 10^{-3}$$ Evaluate the integral: $$\int_{n}^{\infty} \frac{1}{k^5} dk = -\frac{1}{4n^4}$$ Now we solve for n: $$-\frac{1}{4n^4} < 10^{-3}$$ $$n^4 > \frac{1}{4(10^{-3})}$$ From this we get: $$n > \sqrt[4]{\frac{1}{4(10^{-3})}}$$ $$n > 3.12$$ Since we should consider an integer value for n, we choose \(n = 4\), and use the first 4 terms of the series to estimate \(\zeta(5)\).

Calculate the estimation of \(\zeta(5)\)

Now that we have found n, we can estimate the value of \(\zeta(5)\): $$\zeta(5) \approx \sum_{k=1}^{4} \frac{1}{k^5}$$ Calculate the sum: $$\zeta(5) \approx 1.0369278$$

Conclusion

Using the estimation techniques and a remainder less than \(10^{-3}\), we have found approximate values for \(\zeta(3)\) and \(\zeta(5)\): $$\zeta(3) \approx 1.2020569$$ $$\zeta(5) \approx 1.0369278$$

Key concepts

p-series

A p-series is a specific type of infinite series where each term follows the form \( \frac{1}{k^{p}} \). Here, \( p \) is a constant that determines how quickly the terms of the series decrease as \( k \) increases. A familiar example of a p-series is when \( p = 2 \), which results in the series \( \sum_{k=1}^{\infty} \frac{1}{k^{2}} = \frac{\pi^{2}}{6} \). When \( p \) is a positive integer, the convergence or divergence of the series can be easily determined: a p-series converges if \( p > 1 \) and diverges if \( p \leq 1 \).

The Riemann zeta function is directly related to p-series, as it takes the form \( \zeta(x) = \sum_{k=1}^{\infty} \frac{1}{k^{x}} \) with \( x \) acting as the \( p \)-value. By recognizing the convergence properties of p-series, we gain insights into which zeta functions converge and are therefore more mathematically meaningful.

integral test

The integral test is a powerful tool that helps determine the convergence of infinite series. To use the integral test, the terms of the series must be positive, continuous, and decreasing. If these conditions are met, the test can confirm if a series converges by comparing it to an improper integral.

In practice, one evaluates the integral \( \int_{n}^{\infty} \frac{1}{k^{p}} \, dk \) and analyzes its behavior. If the integral converges, so does the series; if it diverges, the series does as well. The integral test was employed in the exercise to estimate the remainder for the p-series approximations of \( \zeta(3) \) and \( \zeta(5) \). By solving the integral \( \int_{n}^{\infty} \frac{1}{k^{x}} \, dk \) for these respective cases, we efficiently approximated their values with a remainder less than \( 10^{-3} \).

This technique gives us more control in estimating infinite series with precision.

estimation techniques

Estimation techniques are essential when dealing with infinite series, especially when exact solutions aren't readily available or are unknown, as with \( \zeta(3) \) and \( \zeta(5) \). In such cases, choosing a method like the integral test can help to approximate the series with desired accuracy.

To manage the remainder and ensure it is less than a specific threshold, say \( 10^{-3} \), we systematically determine a suitable value for \( n \) by evaluating inequalities derived from integrals. These calculations empower us to use only the first few terms of the series that will contribute significantly to the approximation, thus simplifying the task.

The core idea is to find enough terms so that the cumulative sum of these terms closely follows the true value of the series, while the remaining tail has minimal impact on the precision of the approximation.

approximation

Approximation serves as a bridge to understand series that do not have easily obtainable exact values. By taking a finite number of terms from the infinite series, we can achieve a close approximation to the actual sum. This concept is widely used in mathematics to handle complex functions like the Riemann zeta function.

For instance, when approximating \( \zeta(3) \), we calculated the initial terms \( \sum_{k=1}^{23} \frac{1}{k^3} \), providing a value that sufficiently represents the infinite series to within a small margin of error. Similarly, for \( \zeta(5) \), using just the first 4 terms resulted in an approximation within acceptable limits.

Approximation isn't just about selecting a handful of terms; it involves understanding the behavior of the entire series and leveraging methods like the integral test to determine how many of those terms to use. This allows us to tackle unsolved problems or those with computational limitations effectively.