Ryan's Working Page: Difference between revisions
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=Attempt to backwards-engineer a Weil-weighted analog for "Zeta"= | =Attempt to backwards-engineer a Weil-weighted analog for "Zeta"= | ||
In Mike's Zeta Function Working Page, we see that zeta can be thought of as a superposition of weighted "cosine accuracy" functions for every unreduced rational: | In [[Mike's Zeta Function Working Page]], we see that [[zeta]] can be thought of as a superposition of weighted "cosine accuracy" functions for every unreduced rational: | ||
<math>\displaystyle | <math>\displaystyle | ||
\left| \zeta(s) \right|^2 = \zeta(2a) + 2 \sum_{n > d} \left[\frac{\cos\left(b\ln\left(\tfrac{n}{d}\right)\right)}{(nd)^{a}}\right]</math> | \left| \zeta(s) \right|^2 = \zeta(2a) + 2 \sum_{n > d} \left[\frac{\cos\left(b\ln\left(\tfrac{n}{d}\right)\right)}{(nd)^{a}}\right]</math> | ||
The cosines are weighted by 1/(nd)<span style="font-size: 11.6999998092651px; vertical-align: super;">a</span>. However, it is of interest to replace n*d with max(n,d)^2 to see if we can derive a Weil-weighted analog of the Zeta function. I will denote this function by f(s). | The cosines are weighted by 1/(nd)<span style="font-size: 11.6999998092651px; vertical-align: super;">a</span>. However, it is of interest to replace n*d with max(n,d)^2 to see if we can derive a [[Weil norm, Tenney-Weil norm, and TWp interval and tuning space|Weil]]-weighted analog of the Zeta function. I will denote this function by f(s). | ||
<math>\displaystyle | <math>\displaystyle | ||