The Riemann zeta function and tuning/Vector's derivation: Difference between revisions
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Now, this is nowhere differentiable, 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 nowhere differentiable, 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: | ||
<nowiki>$$ \mu_{ | <nowiki>$$ \mu_{b} \left( x \right) = \sum_{k=1}^{\infty}\frac{\cos\left(\log_{2}\left(k\right)\tau x\right)}{k^{2}} $$</nowiki> | ||
Let's clean up the function by removing the scale factors on x: | |||
<nowiki>$$ \mu_{c} \left( x \right) = \sum_{k=1}^{\infty}\frac{\cos x}{k^{2}} $$</nowiki> | |||