Riemann Zeros and Harmonic Frequencies

In summary, the Riemann zeros and primes have a duality relationship, known as the explicit formulae, where the zeros of the Riemann zeta-function can be seen as the harmonic frequencies in the distribution of primes. This is supported by the fact that sums over the complex zeros of the zeta function can be identified with other sums over prime numbers. This concept is further explained through various mathematical explanations and graphical representations.
  • #1
overlook1977
11
0
Can someone elaborate on the relationship of the Riemann zeros and primes? How are the zeros harmonic to the primes? The quotes below mention the 'sum of its complex zeros' and 'other sums over prime numbers'. Can someone clarify this?

From Answers.com:

"The zeros of the Riemann zeta-function and the prime numbers satisfy a certain duality property, known as the explicit formulae, which shows that in the language of Fourier analysis the zeros of the Riemann zeta-function can be regarded as the harmonic frequencies in the distribution of primes."

"In mathematics, the explicit formulae...first case known was for the Riemann zeta function, where sums over its complex zeroes are identified with other sums over prime numbers."
 
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  • #2
I don´t know how deep in the maths you want to go, but you will find a good explanation on this website (with graphics) :

http://secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/encoding2.htm

basically, you will see that Chebyshev's logarithmic prime counting function (which counts not only the primes (2,3,5,7...) but also the prime powers (4,8,9,16,25,...) can be decomposed into the sum of a smooth function
x-ln(2PI)-1/2ln(1-1/x^2) and an infinite sum of logarithmically rescaled siusoids whose frequencies are the imaginary parts of the zeroes of Riemann's zeta function (14.13, 37.58, ...) (assuming the Riemann hypothesis is true, ie the zeroes all have real part 1/2 !).

If you wish to go more in detail check also :

http://secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/berry.htm

which explains a bit more the sentence "primes have music in them..."

And then for a more graphical explanation, with some musical simulation of the frequencies mentionned above, check this website :

http://www.math.ucsb.edu/~stopple/explicit.html


Hope that helps...
 
  • #3


The Riemann zeta-function is a mathematical function that is closely related to the distribution of prime numbers. The Riemann hypothesis, one of the most famous unsolved problems in mathematics, states that all non-trivial zeros of the Riemann zeta-function lie on the critical line Re(s) = 1/2. This means that the zeros of the Riemann zeta-function are closely related to the primes, as they lie on the same line in the complex plane.

The explicit formulae mentioned in the quotes refer to a mathematical relationship between the Riemann zeta-function and the primes. This formula shows that the zeros of the Riemann zeta-function can be seen as the "harmonic frequencies" in the distribution of primes. This means that the position and properties of the Riemann zeros can give us information about the distribution of primes.

For example, the explicit formulae allow us to calculate the number of primes below a certain value by summing over the zeros of the Riemann zeta-function. This is what is meant by "sums over its complex zeros are identified with other sums over prime numbers". In other words, the Riemann zeros and the primes are intricately linked through this explicit formula.

In conclusion, the Riemann zeros and primes have a close relationship through the explicit formulae, which shows that the zeros of the Riemann zeta-function can be seen as the "harmonic frequencies" in the distribution of primes. This allows us to gain insights into the distribution of primes by studying the Riemann zeros, and vice versa. However, the exact nature of this relationship is still a subject of ongoing research.
 

1. What are Riemann Zeros and Harmonic Frequencies?

Riemann Zeros are the values of the complex variable s for which the Riemann zeta function is equal to zero. Harmonic Frequencies are the frequencies at which the vibrations of a system are most stable and have the least amount of energy loss.

2. Why are Riemann Zeros and Harmonic Frequencies important?

Riemann Zeros have a connection to prime numbers and the distribution of prime numbers. Harmonic Frequencies are important in understanding and predicting the behavior of vibrating systems, which can have applications in various fields such as engineering and physics.

3. How are Riemann Zeros and Harmonic Frequencies related?

The Riemann zeta function can be expressed in terms of harmonic frequencies, and the locations of Riemann Zeros on the complex plane have connections to the behavior of certain types of vibrating systems.

4. What is the significance of the Riemann Hypothesis in relation to Riemann Zeros and Harmonic Frequencies?

The Riemann Hypothesis, which states that all non-trivial Riemann Zeros lie on the critical line with a real part of 1/2, has important implications for the distribution of prime numbers and the behavior of harmonic frequencies in vibrating systems.

5. How are Riemann Zeros and Harmonic Frequencies relevant in modern research and technology?

Research on Riemann Zeros and Harmonic Frequencies has led to advancements in fields such as number theory, physics, and engineering. They have also been studied in relation to cryptography and signal processing, making them relevant in modern technology.

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