The Riemann zeta function and tuning/Vector's derivation: Difference between revisions
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[https://www.desmos.com/calculator/6388kalfmq $$ \mu_{c}\left(\sigma, x\right)=\sum_{k=1}^{\infty}\operatorname{Re}\left(k^{ix}\right)k^{-\sigma} $$] | [https://www.desmos.com/calculator/6388kalfmq $$ \mu_{c}\left(\sigma, x\right)=\sum_{k=1}^{\infty}\operatorname{Re}\left(k^{ix}\right)k^{-\sigma} $$] | ||
k<sup>-σ</sup> is a real number, so its real part is equal to itself. Thus | k<sup>-σ</sup> is a real number, so its real part is equal to itself. Thus we can simplify this as follows: | ||
[https://www.desmos.com/calculator/l3q2dtd6xn $$ \mu_{c}\left(\sigma, x\right)=\sum_{k=1}^{\infty}\operatorname{Re}\left(k^{-\sigma+ix}\right) $$] | [https://www.desmos.com/calculator/l3q2dtd6xn $$ \mu_{c}\left(\sigma, x\right)=\sum_{k=1}^{\infty}\operatorname{Re}\left(k^{-\sigma+ix}\right) $$] | ||