The Riemann zeta function and tuning/Vector's derivation: Difference between revisions
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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: | ||
[https://www.desmos.com/calculator/deafikrhvg <nowiki>$$ \mu_{ | [https://www.desmos.com/calculator/deafikrhvg <nowiki>$$ \mu_{a} \left(\sigma, x \right) = \sum_{k=1}^{\infty}\frac{\cos\left(\log_{2}\left(k\right)\tau x\right)}{k^{\sigma}} $$</nowiki>] | ||
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Let's clean up the function by removing the scale factors on x. This just scales the function's inputs from EDO to [[Zetave|EDZ]], and these can be added back later to go back to EDO. | Let's clean up the function by removing the scale factors on x. This just scales the function's inputs from EDO to [[Zetave|EDZ]], and these can be added back later to go back to EDO. | ||
[https://www.desmos.com/calculator/26ypbwbglg <nowiki>$$ \mu_{ | [https://www.desmos.com/calculator/26ypbwbglg <nowiki>$$ \mu_{b} \left(\sigma, x \right) = \sum_{k=1}^{\infty}\frac{\cos (\ln\left(k\right)x)}{k^{\sigma}} $$</nowiki>] | ||