The Riemann zeta function and tuning: Difference between revisions
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The Dirichlet series for the zeta function is absolutely convergent when s>1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2πix/ln(2) corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying [[Wikipedia:Euler summation|Euler summation]]. | The Dirichlet series for the zeta function is absolutely convergent when s>1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2πix/ln(2) corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying [[Wikipedia:Euler summation|Euler summation]]. | ||
== Open problems == | |||
# Can a similar metric be found for edos' performance at approximating arbitrary [[delta-rational]] chords? | |||
== Links == | == Links == | ||
* [https://arxiv.org/abs/math/0309433 X-Ray of Riemann zeta-function] by Juan Arias-de-Reyna | * [https://arxiv.org/abs/math/0309433 X-Ray of Riemann zeta-function] by Juan Arias-de-Reyna | ||