Trivial zeros of the Riemann zeta function

In summary, the Riemann zeta function is a mathematical function named after Bernhard Riemann and denoted by the symbol ζ(s). Its trivial zeros are located at negative even integers and are not as significant as the non-trivial zeros. The non-trivial zeros are closely related to the distribution of prime numbers and their calculation involves various methods, but an explicit calculation method is not known. The Riemann zeta function is also connected to the prime number theorem and has applications in various fields, but its full potential is yet to be explored.
  • #1
mrbohn1
97
0
Clearly I am missing something obvious here, but how is it that negative even numbers are zeros of the Riemann zeta function?

For example:

[tex]\zeta (-2)=1+\frac{1}{2^{-2}}+\frac{1}{3^{-2}}+...=1+4+9+..[/tex]

Which is clearly not zero. What is it that I am doing wrong?
 
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  • #2
You are using the definition of zeta(s) for Re(s) > 1 with a number that has a real part smaller than or equal to 1.
 
  • #3
mrbohn1 said:
What is it that I am doing wrong?

You need not the function you posted, but its analytic continuation.
 
  • #4
Thanks! It all becomes clear.
 
  • #5


The Riemann zeta function is a mathematical function that is widely used in number theory and has many interesting properties. One of these properties is that it has infinitely many zeros, known as the trivial zeros, which lie on the negative even numbers on the complex plane.

It is important to note that the Riemann zeta function is defined for complex numbers, not just real numbers. This means that the input can be a combination of both real and imaginary numbers. When we plug in a negative even number into the zeta function, we are actually plugging in a complex number with an imaginary component of zero. This is because the negative even numbers can be written as -2n, where n is a positive integer. Therefore, when we plug in -2n into the zeta function, we are actually calculating \zeta (-2n) = \sum_{k=1}^\infty k^{-2n} which simplifies to \sum_{k=1}^\infty \frac{1}{k^{2n}}.

Now, let's look at what happens when n = 1. In this case, we have \zeta (-2) = \sum_{k=1}^\infty \frac{1}{k^{2}} = 1+\frac{1}{4}+\frac{1}{9}+... which is the same series that you have written in your question. However, this series does not converge to zero, but rather to a finite value of approximately 1.645. This is because the Riemann zeta function has a special property called analytic continuation, which allows it to be extended to values outside of its original domain. In this case, the zeta function is extended to the negative even numbers on the complex plane, and the series converges to a finite value at these points.

Therefore, it is not that you are doing something wrong, but rather that the Riemann zeta function has unique and interesting properties that may seem counterintuitive at first. I hope this explanation helps to clarify why the negative even numbers are considered to be trivial zeros of the Riemann zeta function.
 

What is the Riemann zeta function and what are its trivial zeros?

The Riemann zeta function is a mathematical function that is defined for all complex numbers except 1. It is named after mathematician Bernhard Riemann and is denoted by the symbol ζ(s). The trivial zeros of the Riemann zeta function are the points where the function equals zero, which are located at negative even integers (-2, -4, -6, etc.). These are called "trivial" because they are easy to find and are not as significant as the non-trivial zeros.

What is the significance of the non-trivial zeros of the Riemann zeta function?

The non-trivial zeros of the Riemann zeta function are closely related to the distribution of prime numbers. In fact, the Riemann hypothesis states that all the non-trivial zeros lie on a specific line in the complex plane, called the critical line. This hypothesis has significant implications in number theory and has been a major open problem in mathematics for over a century.

How are the non-trivial zeros of the Riemann zeta function calculated?

The Riemann zeta function can be calculated using various methods, such as the Euler-Maclaurin formula or the functional equation. However, there is no known method to explicitly calculate the non-trivial zeros. They are usually approximated using numerical methods, such as the Riemann-Siegel formula or the Taylor series expansion.

What is the connection between the Riemann zeta function and the prime number theorem?

The prime number theorem is a fundamental result in number theory that gives an estimate for the number of prime numbers below a given number. It is closely related to the Riemann zeta function through the Riemann-von Mangoldt formula, which expresses the number of primes as a sum involving the zeta function. In fact, the Riemann hypothesis, if proven true, would imply an asymptotic form of the prime number theorem.

Are there any real-world applications of the Riemann zeta function and its trivial zeros?

The Riemann zeta function and its trivial zeros have applications in various fields, including physics, engineering, and cryptography. For example, the Riemann hypothesis has been used to improve the efficiency of algorithms in computer science, and the zeta function has been used in the study of quantum chaos. However, the full potential of the Riemann zeta function and its zeros is yet to be explored.

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