Ryan's Working Page: Difference between revisions

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=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

\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

@@ Line 1: / Line 1: @@
 =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
 \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