The Riemann zeta function and tuning/Vector's derivation: Difference between revisions
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Let's clean up the function by removing the scale factors on x: | Let's clean up the function by removing the scale factors on x: | ||
<nowiki>$$ \mu_{c} \left( x \right) = \sum_{k=1}^{\infty}\frac{\cos (\ | <nowiki>$$ \mu_{c} \left( x \right) = \sum_{k=1}^{\infty}\frac{\cos (\ln\left(k\right)x)}{k^{2}} $$</nowiki> | ||
By the complex exponential theorem, we know that | |||
$$ e^{ix}=\cos\left(x\right)+i\sin\left(x\right) $$ | |||
so that cos(x) can be rewritten as Re(e<sup>ix</sup>). | |||
<nowiki>$$ \mu_{d}\left(x\right)=\sum_{k=1}^{infty}\frac{\operatorname{Re}\left(e^{i\left(\ln\left(k\right)x\right)}\right)}{k^{2}} $$</nowiki> | |||
For now, we will ignore the Re() function as a sum of real parts is the same as the real part of the sum (by the rules of complex addition), and the denominator is just a real number. | |||
<nowiki>$$ \mu_{d}\left(x\right)=\sum_{k=1}^{infty}\frac{e^{i\left(\ln\left(k\right)x\right)}}{k^{2}} $$</nowiki> | |||