The Riemann zeta function and tuning: Difference between revisions
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This new function has the property that {{nowrap|F<sub>s</sub>(''x'') {{=}} F<sub>s</sub>(0) − E<sub>s</sub>(''x'')}}, so that all we have done is flip the sign of E<sub>s</sub>(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum. | This new function has the property that {{nowrap|F<sub>s</sub>(''x'') {{=}} F<sub>s</sub>(0) − E<sub>s</sub>(''x'')}}, so that all we have done is flip the sign of E<sub>s</sub>(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum. | ||
Of more interest is the fact that it is a known mathematical function. | |||
The logarithm of the {{w|Riemann zeta function}} function can be expressed in terms of a {{w|Dirichlet series}} involving the von Mangoldt function: | Of more interest is the fact that it is a known mathematical function. The logarithm of the {{w|Riemann zeta function}} function [[The Riemann zeta function and tuning/Appendix#1a. Dirichlet series for the von Mangoldt function|can be expressed]] in terms of a {{w|Dirichlet series}} involving the von Mangoldt function: | ||
<math>\displaystyle | <math>\displaystyle | ||