The Riemann zeta function and tuning/Vector's derivation

We start with the generalized mu function, which sums up the relative error on all integer harmonics weighted by an inverse power of the harmonic:

$$ \mu \left(\sigma, x \right) = \sum_{k=1}^{\infty} \frac{\operatorname{abs} \left( \operatorname{mod} \left( 2\log_{2} \left( k \right) x, 2 \right) - 1 \right)}{k^{\sigma}} $$

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:

$$ \mu_{b} \left(\sigma, x \right) = \sum_{k=1}^{\infty}\frac{\cos\left(\log_{2}\left(k\right)\tau x\right)}{k^{\sigma}} $$

Let's clean up the function by removing the scale factors on x:

$$ \mu_{c} \left(\sigma, x \right) = \sum_{k=1}^{\infty}\frac{\cos (\ln\left(k\right)x)}{k^{\sigma}} $$

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^ix).

$$ \mu_{c}\left(\sigma, x\right)=\sum_{k=1}^{\infty}\frac{\operatorname{Re}\left(e^{i\left(\ln\left(k\right)x\right)}\right)}{k^{\sigma}} $$

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.

$$ \mu_{d}\left(\sigma, x\right)=\sum_{k=1}^{\infty}\frac{e^{i\left(\ln\left(k\right)x\right)}}{k^{\sigma}} $$

e^ln(n)x = n^x, so:

$$ \mu_{d}\left(\sigma, x\right)=\sum_{k=1}^{\infty}\frac{k^{ix}}{k^{\sigma}} $$

Thus:

$$ \mu_{d}\left(\sigma, x\right)=\sum_{k=1}^{\infty}k^{ix}k^{-\sigma} $$

so

$$ \mu_{d}\left(\sigma, x\right)=\sum_{k=1}^{\infty}k^{-\sigma+ix} $$

-σ+ix is just a complex number, which we may write as -s:

$$ \mu_{e}\left(s\right)=\sum_{k=1}^{\infty}k^{-s} $$

But this is the definition of the Riemann zeta function, so μ_e(s) = ζ(s), and re-adding the Re() function gives Re(ζ(s)) with s = σ-ix; x is the equal division and σ is the weight.