The Riemann zeta function and tuning: Difference between revisions
relationship to HE |
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<math>\xi(x) = \sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2</math> | <math>\xi(x) = \sum_2^p (\frac{||x \log_2 q||}{\log_2 q})^2</math> | ||
This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of x which are the [[Tenney-Euclidean_Tuning|Tenney-Euclidean tuning]]s of the octaves of the associated vals, while ξ for these minima is the square of the [[Tenney-Euclidean_metrics|Tenney-Euclidean relative error]] of the val. | This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of x which are the [[Tenney-Euclidean_Tuning|Tenney-Euclidean tuning]]s of the octaves of the associated vals, while ξ for these minima is the square of the [[Tenney-Euclidean_metrics|Tenney-Euclidean relative error]] of the val - equal to the TE error times the TE complexity, and sometimes known as "TE simple badness." | ||
Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could change the weighting factor to a power so that it does converge: | Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could change the weighting factor to a power so that it does converge: | ||