In summary, Euler developed some remarkable mathematical formulas, including the famous identity ##e^{i \pi} = -1##. One of his original contributions is the equation ##1 + 1/4 + 1/9 + 1/16 + ... = \pi^2/6##, which can also be written as ##\zeta(2) = \pi^2/6## using the Riemann Zeta function. This function is defined by ##\zeta(s) = \sum_{j=1}^\infty \dfrac{1}{j^s}## when ##s > 1##. Euler's identity can also be expressed as ##F(x) = \sum_{
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Euler’s amazing identity
The mathematician Leonard Euler developed some surprising mathematical formulas involving the number ##\pi##. The most famous equation is ##e^{i \pi} = -1##, which is one of the most important equations in modern mathematics, but unfortunately, it wasn’t invented by Euler.Something that is original with Euler is this amazing identity:
Equation 1: ##1 + 1/4 + 1/9 + 1/16 + … = \pi^2/6##
This is one instance of an important function called the Riemann Zeta function, ##\zeta(s)##, which in the case where ##s > 1## is defined by:
Equation 2: ##\zeta(s) = \sum_{j=1}^\infty \dfrac{1}{j^s}##
So Euler’s identity can be written as:
Equation 3: ##\zeta(2) = \frac{\pi^2}{6}##
This post is an attempt to show how you can derive that result, and related results, using facts about trigonometry, complex numbers, and the Fourier series.
Some related functions defined...

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Just below the heading "Equation-7". Also same identity below the heading "Equation-9" and before "Sum-1".
##F(x)= \sum_{j=-\infty}^{\infty} e^{ijx}=\sum_{j=-\infty}^{-1} e^{ijx}+1+\sum_{j=0}^{\infty} e^{ijx} ##

Shouldn't it be(?):
##F(x)= \sum_{j=-\infty}^{\infty} e^{ijx}=\sum_{j=-\infty}^{-1} e^{ijx}+1+\sum_{j=1}^{\infty} e^{ijx} ##
 
  • #6
Paul Colby said:
I've checked chrome and safari and your link is broken in both. There appears to be garbage prior to the working URL. Is this the correct one?
Yes. I have extended the results in that insight and I will update it Really Soon Now (as Jerry Pournelle used to say).
 

What is the Riemann Zeta Function?

The Riemann Zeta Function is a mathematical function that was first introduced by the mathematician Bernhard Riemann. It is defined as the infinite sum of the reciprocal of the powers of all positive integers. In other words, it is represented by the formula ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ...

Why is computing the Riemann Zeta Function important?

The Riemann Zeta Function is important in many areas of mathematics, including number theory, analysis, and physics. It has connections to prime numbers, the distribution of primes, and the behavior of complex numbers. It also plays a crucial role in the study of the distribution of prime numbers and the Riemann Hypothesis, one of the most famous unsolved problems in mathematics.

What is the Fourier Series method for computing the Riemann Zeta Function?

The Fourier Series method is a mathematical technique for representing a function as an infinite sum of trigonometric functions. In the case of the Riemann Zeta Function, it involves expressing the function as a sum of sines and cosines, which can then be used to approximate the value of the function at any point.

What are the advantages of using Fourier Series to compute the Riemann Zeta Function?

One of the main advantages of using Fourier Series is that it allows for a more efficient and accurate computation of the Riemann Zeta Function compared to other methods. This is because the Fourier Series method takes advantage of the periodicity and symmetry of the function, which leads to faster convergence and better approximations.

Are there any limitations to using Fourier Series to compute the Riemann Zeta Function?

While the Fourier Series method is a powerful tool for computing the Riemann Zeta Function, it does have some limitations. One of the main limitations is that it can only be used to compute the function for values of s greater than 1. Additionally, the method requires a large number of terms in the series to achieve high levels of accuracy, which can be computationally expensive.

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