Harmonic entropy: Difference between revisions
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$$\displaystyle |\zeta(w+i t)|^2 = \left[ \sum_{j_c \in \mathbb{N}^+} \frac{1}{{j_c}^{2w}} \right] \cdot \left[ \sum_{j \in \mathbb{Q}} \frac{e^{i t \log (\frac{j_{n'}}{j_{d'}})}}{(j_{n'} j_{d'})^{w}} \right]$$ | $$\displaystyle |\zeta(w+i t)|^2 = \left[ \sum_{j_c \in \mathbb{N}^+} \frac{1}{{j_c}^{2w}} \right] \cdot \left[ \sum_{j \in \mathbb{Q}} \frac{e^{i t \log (\frac{j_{n'}}{j_{d'}})}}{(j_{n'} j_{d'})^{w}} \right]$$ | ||
where the left summation now has {{nowrap|''j''<sub>''c''</sub> ∈ ℕ{{ | where the left summation now has {{nowrap|''j''<sub>''c''</sub> ∈ ℕ{{+}}}}, the set of strictly positive rational numbers, and the right summation now has {{nowrap|''j'' ∈ ℚ}} the set of reduced rationals. Note again that the product above yields all unreduced rationals, thanks to the ''j''<sub>''c''</sub>. | ||
Now, note that that left series is, itself, just another Dirichlet series that converges to the zeta function. We have | Now, note that that left series is, itself, just another Dirichlet series that converges to the zeta function. We have | ||