Riemann Zeros and Harmonic Frequencies

[overlook1977]

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."

 

[chrisina]

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...