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