The Riemann zeta function and tuning/Vector's derivation: Difference between revisions
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We start with the generalized mu function, which sums up the relative error on all integer harmonics weighted by an inverse power of the harmonic: | We start with the generalized mu function, which sums up the relative error on all integer harmonics weighted by an inverse power of the harmonic: | ||
[https://www.desmos.com/calculator/4zcynoue8s <nowiki>$$ \mu \left(\sigma, x \right) = \sum_{k=1}^{\infty} \frac{\ | [https://www.desmos.com/calculator/4zcynoue8s <nowiki>$$ \mu \left(\sigma, x \right) = \sum_{k=1}^{\infty} \frac{\abs{ \operatorname{mod} \left( 2\log_{2} \left( k \right) x, 2 \right) - 1 }}{k^{\sigma}} $$</nowiki>] | ||
Now, this is a rather annoying function to work with for math reasons, so it might be useful to replace the "zigzag" that we use as our error function with a "smoother" alternative. The most obvious answer is cosine: | Now, this is a rather annoying function to work with for math reasons, so it might be useful to replace the "zigzag" that we use as our error function with a "smoother" alternative. The most obvious answer is cosine: | ||