User:Sintel/Zeta working page: Difference between revisions
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We can remove a finite list of primes from consideration by multiplying <math>\zeta(s)</math> by the corresponding factors <math>(1 - p^{-s})</math> for each prime ''p'' we wish to remove. | We can remove a finite list of primes from consideration by multiplying <math>\zeta(s)</math> by the corresponding factors <math>(1 - p^{-s})</math> for each prime ''p'' we wish to remove. | ||
After doing this, the smallest prime remaining will dominate peak values for ''s''. | After doing this, the smallest prime remaining will dominate peak values for ''s''. | ||
In particular if we remove the prime 2, <math>\left(1 - 2^{- | In particular if we remove the prime 2, <math>\left(1 - 2^{-s}\right)\zeta(s)</math> is now dominated by 3, and the large peak values occur near [[edt|equal divisions of the tritave]] (3/1). | ||
Along any line of constant <math>\sigma</math>, [[The Riemann zeta function and tuning/Appendix#1b. Conversion factor for removing primes|it can be shown that]]: | Along any line of constant <math>\sigma</math>, [[The Riemann zeta function and tuning/Appendix#1b. Conversion factor for removing primes|it can be shown that]]: | ||