- it's not a Fourier series, but it is a Fourier transform if we replace the variable x by u = ln(x). Then each oscillation in Riemann's explicit formula is a constant times a sine wave whose frequency is the imaginary part of the Riemann zeta zero. (I'm oversimplifying a bit, but not in a bad way.) So, we are taking the Fourier transform of a sum of constants times Dirac deltas.
In short, the connection is VERY deep. This is why people talk about "the music of the primes" when discussing the Riemann zeta function!
https://en.wikipedia.org/wiki/The_Music_of_the_Primes
John Carlos Baez
npub1nf…3nqe4
2026-10-28 11:25:43 UTC
in reply to nevent1q…e54m
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2026-10-28 11:25:43 UTCEvent JSON
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"content": "- it's not a Fourier series, but it is a Fourier transform if we replace the variable x by u = ln(x). Then each oscillation in Riemann's explicit formula is a constant times a sine wave whose frequency is the imaginary part of the Riemann zeta zero. (I'm oversimplifying a bit, but not in a bad way.) So, we are taking the Fourier transform of a sum of constants times Dirac deltas. \n\nIn short, the connection is VERY deep. This is why people talk about \"the music of the primes\" when discussing the Riemann zeta function!\n\nhttps://en.wikipedia.org/wiki/The_Music_of_the_Primes",
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